1-5 Parent Functions and Transformations

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1-5 Assignment - Parent Functions and Transformations 1-5 Bell Work - Parent Functions and Transformations 1-5 Exit Quiz - Parent Functions and Transformations 1-5 Guided Notes SE - Parent Functions and Transformations 1-5 Guided Notes TE - Parent Functions and Transformations 1-5 Lesson Plan - Parent Functions and Transformations 1-5 Online Activities - Parent Functions and Transformations 1-5 Slide Show - Parent Functions and Transformations
1-5 Assignment - Parent Functions and Transformations 1-5 Bell Work - Parent Functions and Transformations 1-5 Exit Quiz - Parent Functions and Transformations 1-5 Guided Notes SE - Parent Functions and Transformations   1-5 Guided Notes TE - Parent Functions and Transformations 1-5 Lesson Plan - Parent Functions and Transformations 1-5 Online Activities - Parent Functions and Transformations 1-5 Slide Show - Parent Functions and Transformations

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pre calc transformations worksheet

Precalculus MAT 206–0801

Spring 2020 BMCC Prof. Dr. Ivan Retamoso

Relations and Functions

Evaluations of Functions

The Horizontal Line Test

Finding the Domain of a Function

Finding the Domain and Range of a Function from its Graph

Piecewise Functions

Average Rate of Change of a Function

Increasing and Decreasing Functions

Relative Maxima or Minima

Arithmetic Operations with Functions

Compositions of Functions

Transformation of Functions

Absolute Value Functions

Inverse Functions

Linear Functions

Graph of Linear Functions

Modeling with Linear Functions

Operations with Complex Numbers

Vertex and Axis of Symmetry of a Parabola

Minimum or Maximum of a Parabola

Polynomial Functions of Higher Degree

Polynomial and Synthetic Division

The Reminder and Factor Theorem

The Rational Zero Theorem

Zeros of a Polynomial Function

Rational Functions

Graphing Rational Functions

Slant Asymptote

Exponential Functions

Logarithmic Functions

Properties of Logarithms

Exponential and Logarithmic Equations

Radian and Degree Measure

Right Triangle Trigonometry

Trigonometric Functions of any Angle

The Unit Circle

Graph of Trigonometric Functions

Inverse Trigonometric Functions

Sum and Difference Identities

Double-Angle and Half-Angle Formulas

Product to Sum and Sum to Product Formulas

Solving Trigonometric Equations

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pre calc transformations worksheet

TRANSFORMATIONS & FUNCTIONS

Chapter 1 - Function Transformations

Chapter 2 - Radical Functions

Chapter 3 - Polynomial Functions

• UNIT 1 NOTES PACKAGE •

1.1 - HORIZONTAL & VERTICAL TRANSLATIONS

PG 12  #1-12

1.2 - REFLECTIONS & STRETCHES

PG 28  #1-4, 6, 7, 9, 14

1.3 - COMBINING TRANSFORMATIONS

TRANSFORMATIONS (handout)

PG 38  #1-11, 15

TRANSFORMATIONS CONT...

ANSWER KEY 

1.4 - INVERSE OF A RELATION

PG 51  #1-10, 12, 15, 20

2.1 - RADICAL FUNCTIONS & TRANSFORMATIONS

PG 72  #1-7, 10, 11

2.2 - SQUARE ROOT OF A FUNCTION

PG 86  #3-11

2.3 - SOLVING RADICAL EQUATIONS GRAPHICALLY

PG 96  #3-9, 14, 16

3.1 - CHARACTERISTICS OF POLYNOMIAL FUNCTIONS

PG 114  #1-4, 6, 9, 11

3.2 (1) - THE REMAINDER THEOREM

PG 124  #1-5

3.2 (2) - THE REMAINDER THEOREM

PG 124  #6-10, 14, 15

3.3 - THE FACTOR THEOREM

PG 133  #3-7

3.3 - CONT...

3.4 (1) - solving polynomial equations, 3.4 ( 2 ) - zero of multiplicity.

PG 147  #1-7, 9, 10

unit 1 test cycle

REVIEW PAGES:

CH 1 - PG 56

CH 2 - PG 99

CH 3 - PG 153

UNIT 1 - PG 158

PRE-TEST - Tuesday, F eb 27th

CORRECTIONS - Wednesday, Feb 28th

UNIT TEST - Thu rsday, Feb 29th (LEAP DAY!!!)

CORRECTIONS - Friday, Mar 1st

RE-TEST - Monday, Mar 4th

  • 1.5 Transformation of Functions
  • Introduction to Functions
  • 1.1 Functions and Function Notation
  • 1.2 Domain and Range
  • 1.3 Rates of Change and Behavior of Graphs
  • 1.4 Composition of Functions
  • 1.6 Absolute Value Functions
  • 1.7 Inverse Functions
  • Key Equations
  • Key Concepts
  • Review Exercises
  • Practice Test
  • Introduction to Linear Functions
  • 2.1 Linear Functions
  • 2.2 Graphs of Linear Functions
  • 2.3 Modeling with Linear Functions
  • 2.4 Fitting Linear Models to Data
  • Introduction to Polynomial and Rational Functions
  • 3.1 Complex Numbers
  • 3.2 Quadratic Functions
  • 3.3 Power Functions and Polynomial Functions
  • 3.4 Graphs of Polynomial Functions
  • 3.5 Dividing Polynomials
  • 3.6 Zeros of Polynomial Functions
  • 3.7 Rational Functions
  • 3.8 Inverses and Radical Functions
  • 3.9 Modeling Using Variation
  • Introduction to Exponential and Logarithmic Functions
  • 4.1 Exponential Functions
  • 4.2 Graphs of Exponential Functions
  • 4.3 Logarithmic Functions
  • 4.4 Graphs of Logarithmic Functions
  • 4.5 Logarithmic Properties
  • 4.6 Exponential and Logarithmic Equations
  • 4.7 Exponential and Logarithmic Models
  • 4.8 Fitting Exponential Models to Data
  • Introduction to Trigonometric Functions
  • 5.2 Unit Circle: Sine and Cosine Functions
  • 5.3 The Other Trigonometric Functions
  • 5.4 Right Triangle Trigonometry
  • Introduction to Periodic Functions
  • 6.1 Graphs of the Sine and Cosine Functions
  • 6.2 Graphs of the Other Trigonometric Functions
  • 6.3 Inverse Trigonometric Functions
  • Introduction to Trigonometric Identities and Equations
  • 7.1 Solving Trigonometric Equations with Identities
  • 7.2 Sum and Difference Identities
  • 7.3 Double-Angle, Half-Angle, and Reduction Formulas
  • 7.4 Sum-to-Product and Product-to-Sum Formulas
  • 7.5 Solving Trigonometric Equations
  • 7.6 Modeling with Trigonometric Functions
  • Introduction to Further Applications of Trigonometry
  • 8.1 Non-right Triangles: Law of Sines
  • 8.2 Non-right Triangles: Law of Cosines
  • 8.3 Polar Coordinates
  • 8.4 Polar Coordinates: Graphs
  • 8.5 Polar Form of Complex Numbers
  • 8.6 Parametric Equations
  • 8.7 Parametric Equations: Graphs
  • 8.8 Vectors
  • Introduction to Systems of Equations and Inequalities
  • 9.1 Systems of Linear Equations: Two Variables
  • 9.2 Systems of Linear Equations: Three Variables
  • 9.3 Systems of Nonlinear Equations and Inequalities: Two Variables
  • 9.4 Partial Fractions
  • 9.5 Matrices and Matrix Operations
  • 9.6 Solving Systems with Gaussian Elimination
  • 9.7 Solving Systems with Inverses
  • 9.8 Solving Systems with Cramer's Rule
  • Introduction to Analytic Geometry
  • 10.1 The Ellipse
  • 10.2 The Hyperbola
  • 10.3 The Parabola
  • 10.4 Rotation of Axes
  • 10.5 Conic Sections in Polar Coordinates
  • Introduction to Sequences, Probability and Counting Theory
  • 11.1 Sequences and Their Notations
  • 11.2 Arithmetic Sequences
  • 11.3 Geometric Sequences
  • 11.4 Series and Their Notations
  • 11.5 Counting Principles
  • 11.6 Binomial Theorem
  • 11.7 Probability
  • Introduction to Calculus
  • 12.1 Finding Limits: Numerical and Graphical Approaches
  • 12.2 Finding Limits: Properties of Limits
  • 12.3 Continuity
  • 12.4 Derivatives
  • A | Basic Functions and Identities

Learning Objectives

In this section, you will:

  • Graph functions using vertical and horizontal shifts.
  • Graph functions using reflections about the x x -axis and the y y -axis.
  • Determine whether a function is even, odd, or neither from its graph.
  • Graph functions using compressions and stretches.
  • Combine transformations.

We all know that a flat mirror enables us to see an accurate image of ourselves and whatever is behind us. When we tilt the mirror, the images we see may shift horizontally or vertically. But what happens when we bend a flexible mirror? Like a carnival funhouse mirror, it presents us with a distorted image of ourselves, stretched or compressed horizontally or vertically. In a similar way, we can distort or transform mathematical functions to better adapt them to describing objects or processes in the real world. In this section, we will take a look at several kinds of transformations.

Graphing Functions Using Vertical and Horizontal Shifts

Often when given a problem, we try to model the scenario using mathematics in the form of words, tables, graphs, and equations. One method we can employ is to adapt the basic graphs of the toolkit functions to build new models for a given scenario. There are systematic ways to alter functions to construct appropriate models for the problems we are trying to solve.

Identifying Vertical Shifts

One simple kind of transformation involves shifting the entire graph of a function up, down, right, or left. The simplest shift is a vertical shift , moving the graph up or down, because this transformation involves adding a positive or negative constant to the function. In other words, we add the same constant to the output value of the function regardless of the input. For a function g ( x ) = f ( x ) + k , g ( x ) = f ( x ) + k , the function f ( x ) f ( x ) is shifted vertically k k units. See Figure 2 for an example.

To help you visualize the concept of a vertical shift, consider that y = f ( x ) . y = f ( x ) . Therefore, f ( x ) + k f ( x ) + k is equivalent to y + k . y + k . Every unit of y y is replaced by y + k , y + k , so the y - y - value increases or decreases depending on the value of k . k . The result is a shift upward or downward.

Vertical Shift

Given a function f ( x ) , f ( x ) , a new function g ( x ) = f ( x ) + k , g ( x ) = f ( x ) + k , where k k is a constant, is a vertical shift of the function f ( x ) . f ( x ) . All the output values change by k k units. If k k is positive, the graph will shift up. If k k is negative, the graph will shift down.

Adding a Constant to a Function

To regulate temperature in a green building, airflow vents near the roof open and close throughout the day. Figure 3 shows the area of open vents V V (in square feet) throughout the day in hours after midnight, t . t . During the summer, the facilities manager decides to try to better regulate temperature by increasing the amount of open vents by 20 square feet throughout the day and night. Sketch a graph of this new function.

We can sketch a graph of this new function by adding 20 to each of the output values of the original function. This will have the effect of shifting the graph vertically up, as shown in Figure 4 .

Notice that in Figure 4 , for each input value, the output value has increased by 20, so if we call the new function S ( t ) , S ( t ) , we could write

This notation tells us that, for any value of t , S ( t ) t , S ( t ) can be found by evaluating the function V V at the same input and then adding 20 to the result. This defines S S as a transformation of the function V , V , in this case a vertical shift up 20 units. Notice that, with a vertical shift, the input values stay the same and only the output values change. See Table 1 .

Given a tabular function, create a new row to represent a vertical shift.

  • Identify the output row or column.
  • Determine the magnitude of the shift.
  • Add the shift to the value in each output cell. Add a positive value for up or a negative value for down.

Shifting a Tabular Function Vertically

A function f ( x ) f ( x ) is given in Table 2 . Create a table for the function g ( x ) = f ( x ) − 3. g ( x ) = f ( x ) − 3.

The formula g ( x ) = f ( x ) − 3 g ( x ) = f ( x ) − 3 tells us that we can find the output values of g g by subtracting 3 from the output values of f . f . For example:

Subtracting 3 from each f ( x ) f ( x ) value, we can complete a table of values for g ( x ) g ( x ) as shown in Table 3 .

As with the earlier vertical shift, notice the input values stay the same and only the output values change.

The function h ( t ) = − 4.9 t 2 + 30 t h ( t ) = − 4.9 t 2 + 30 t gives the height h h of a ball (in meters) thrown upward from the ground after t t seconds. Suppose the ball was instead thrown from the top of a 10-m building. Relate this new height function b ( t ) b ( t ) to h ( t ) , h ( t ) , and then find a formula for b ( t ) . b ( t ) .

Identifying Horizontal Shifts

We just saw that the vertical shift is a change to the output, or outside, of the function. We will now look at how changes to input, on the inside of the function, change its graph and meaning. A shift to the input results in a movement of the graph of the function left or right in what is known as a horizontal shift , shown in Figure 5 .

For example, if f ( x ) = x 2 , f ( x ) = x 2 , then g ( x ) = ( x − 2 ) 2 g ( x ) = ( x − 2 ) 2 is a new function. Each input is reduced by 2 prior to squaring the function. The result is that the graph is shifted 2 units to the right, because we would need to increase the prior input by 2 units to yield the same output value as given in f . f .

Horizontal Shift

Given a function f , f , a new function g ( x ) = f ( x − h ) , g ( x ) = f ( x − h ) , where h h is a constant, is a horizontal shift of the function f . f . If h h is positive, the graph will shift right. If h h is negative, the graph will shift left.

Adding a Constant to an Input

Returning to our building airflow example from Figure 3 , suppose that in autumn the facilities manager decides that the original venting plan starts too late, and wants to begin the entire venting program 2 hours earlier. Sketch a graph of the new function.

We can set V ( t ) V ( t ) to be the original program and F ( t ) F ( t ) to be the revised program.

In the new graph, at each time, the airflow is the same as the original function V V was 2 hours later. For example, in the original function V , V , the airflow starts to change at 8 a.m., whereas for the function F , F , the airflow starts to change at 6 a.m. The comparable function values are V ( 8 ) = F ( 6 ) . V ( 8 ) = F ( 6 ) . See Figure 6 . Notice also that the vents first opened to 220  ft 2 220  ft 2 at 10 a.m. under the original plan, while under the new plan the vents reach 220  ft 2 220  ft 2 at 8 a.m., so V ( 10 ) = F ( 8 ) . V ( 10 ) = F ( 8 ) .

In both cases, we see that, because F ( t ) F ( t ) starts 2 hours sooner, h = − 2. h = − 2. That means that the same output values are reached when F ( t ) = V ( t − ( − 2 ) ) = V ( t + 2 ) . F ( t ) = V ( t − ( − 2 ) ) = V ( t + 2 ) .

Note that V ( t + 2 ) V ( t + 2 ) has the effect of shifting the graph to the left .

Horizontal changes or “inside changes” affect the domain of a function (the input) instead of the range and often seem counterintuitive. The new function F ( t ) F ( t ) uses the same outputs as V ( t ) , V ( t ) , but matches those outputs to inputs 2 hours earlier than those of V ( t ) . V ( t ) . Said another way, we must add 2 hours to the input of V V to find the corresponding output for F : F ( t ) = V ( t + 2 ) . F : F ( t ) = V ( t + 2 ) .

Given a tabular function, create a new row to represent a horizontal shift.

  • Identify the input row or column.
  • Add the shift to the value in each input cell.

Shifting a Tabular Function Horizontally

A function f ( x ) f ( x ) is given in Table 4 . Create a table for the function g ( x ) = f ( x − 3 ) . g ( x ) = f ( x − 3 ) .

The formula g ( x ) = f ( x − 3 ) g ( x ) = f ( x − 3 ) tells us that the output values of g g are the same as the output value of f f when the input value is 3 less than the original value. For example, we know that f ( 2 ) = 1. f ( 2 ) = 1. To get the same output from the function g , g , we will need an input value that is 3 larger . We input a value that is 3 larger for g ( x ) g ( x ) because the function takes 3 away before evaluating the function f . f .

We continue with the other values to create Table 5 .

The result is that the function g ( x ) g ( x ) has been shifted to the right by 3. Notice the output values for g ( x ) g ( x ) remain the same as the output values for f ( x ) , f ( x ) , but the corresponding input values, x , x , have shifted to the right by 3. Specifically, 2 shifted to 5, 4 shifted to 7, 6 shifted to 9, and 8 shifted to 11.

Figure 7 represents both of the functions. We can see the horizontal shift in each point.

Identifying a Horizontal Shift of a Toolkit Function

Figure 8 represents a transformation of the toolkit function f ( x ) = x 2 . f ( x ) = x 2 . Relate this new function g ( x ) g ( x ) to f ( x ) , f ( x ) , and then find a formula for g ( x ) . g ( x ) .

Notice that the graph is identical in shape to the f ( x ) = x 2 f ( x ) = x 2 function, but the x- values are shifted to the right 2 units. The vertex used to be at (0,0), but now the vertex is at (2,0). The graph is the basic quadratic function shifted 2 units to the right, so

Notice how we must input the value x = 2 x = 2 to get the output value y = 0 ; y = 0 ; the x -values must be 2 units larger because of the shift to the right by 2 units. We can then use the definition of the f ( x ) f ( x ) function to write a formula for g ( x ) g ( x ) by evaluating f ( x − 2 ) . f ( x − 2 ) .

To determine whether the shift is + 2 + 2 or − 2 − 2 , consider a single reference point on the graph. For a quadratic, looking at the vertex point is convenient. In the original function, f ( 0 ) = 0. f ( 0 ) = 0. In our shifted function, g ( 2 ) = 0. g ( 2 ) = 0. To obtain the output value of 0 from the function f , f , we need to decide whether a plus or a minus sign will work to satisfy g ( 2 ) = f ( x − 2 ) = f ( 0 ) = 0. g ( 2 ) = f ( x − 2 ) = f ( 0 ) = 0. For this to work, we will need to subtract 2 units from our input values.

Interpreting Horizontal versus Vertical Shifts

The function G ( m ) G ( m ) gives the number of gallons of gas required to drive m m miles. Interpret G ( m ) + 10 G ( m ) + 10 and G ( m + 10 ) . G ( m + 10 ) .

G ( m ) + 10 G ( m ) + 10 can be interpreted as adding 10 to the output, gallons. This is the gas required to drive m m miles, plus another 10 gallons of gas. The graph would indicate a vertical shift.

G ( m + 10 ) G ( m + 10 ) can be interpreted as adding 10 to the input, miles. So this is the number of gallons of gas required to drive 10 miles more than m m miles. The graph would indicate a horizontal shift.

Given the function f ( x ) = x , f ( x ) = x , graph the original function f ( x ) f ( x ) and the transformation g ( x ) = f ( x + 2 ) g ( x ) = f ( x + 2 ) on the same axes. Is this a horizontal or a vertical shift? Which way is the graph shifted and by how many units?

Combining Vertical and Horizontal Shifts

Now that we have two transformations, we can combine them together. Vertical shifts are outside changes that affect the output ( y - y - ) axis values and shift the function up or down. Horizontal shifts are inside changes that affect the input ( x - x - ) axis values and shift the function left or right. Combining the two types of shifts will cause the graph of a function to shift up or down and right or left.

Given a function and both a vertical and a horizontal shift, sketch the graph.

  • Identify the vertical and horizontal shifts from the formula.
  • The vertical shift results from a constant added to the output. Move the graph up for a positive constant and down for a negative constant.
  • The horizontal shift results from a constant added to the input. Move the graph left for a positive constant and right for a negative constant.
  • Apply the shifts to the graph in either order.

Graphing Combined Vertical and Horizontal Shifts

Given f ( x ) = | x | , f ( x ) = | x | , sketch a graph of h ( x ) = f ( x + 1 ) − 3. h ( x ) = f ( x + 1 ) − 3.

The function f f is our toolkit absolute value function. We know that this graph has a V shape, with the point at the origin. The graph of h h has transformed f f in two ways: f ( x + 1 ) f ( x + 1 ) is a change on the inside of the function, giving a horizontal shift left by 1, and the subtraction by 3 in f ( x + 1 ) − 3 f ( x + 1 ) − 3 is a change to the outside of the function, giving a vertical shift down by 3. The transformation of the graph is illustrated in Figure 9 .

Let us follow one point of the graph of f ( x ) = | x | . f ( x ) = | x | .

  • The point ( 0 , 0 ) ( 0 , 0 ) is transformed first by shifting left 1 unit: ( 0 , 0 ) → ( −1 , 0 ) ( 0 , 0 ) → ( −1 , 0 )
  • The point ( −1 , 0 ) ( −1 , 0 ) is transformed next by shifting down 3 units: ( −1 , 0 ) → ( −1 , −3 ) ( −1 , 0 ) → ( −1 , −3 )

Figure 10 shows the graph of h . h .

Given f ( x ) = | x | , f ( x ) = | x | , sketch a graph of h ( x ) = f ( x − 2 ) + 4. h ( x ) = f ( x − 2 ) + 4.

Identifying Combined Vertical and Horizontal Shifts

Write a formula for the graph shown in Figure 11 , which is a transformation of the toolkit square root function.

The graph of the toolkit function starts at the origin, so this graph has been shifted 1 to the right and up 2. In function notation, we could write that as

Using the formula for the square root function, we can write

Note that this transformation has changed the domain and range of the function. This new graph has domain [ 1 , ∞ ) [ 1 , ∞ ) and range [ 2 , ∞ ) . [ 2 , ∞ ) .

Write a formula for a transformation of the toolkit reciprocal function f ( x ) = 1 x f ( x ) = 1 x that shifts the function’s graph one unit to the right and one unit up.

Graphing Functions Using Reflections about the Axes

Another transformation that can be applied to a function is a reflection over the x - or y -axis. A vertical reflection reflects a graph vertically across the x -axis, while a horizontal reflection reflects a graph horizontally across the y -axis. The reflections are shown in Figure 12 .

Notice that the vertical reflection produces a new graph that is a mirror image of the base or original graph about the x -axis. The horizontal reflection produces a new graph that is a mirror image of the base or original graph about the y -axis.

Reflections

Given a function f ( x ), f ( x ), a new function g ( x ) = − f ( x ) g ( x ) = − f ( x ) is a vertical reflection of the function f ( x ) , f ( x ) , sometimes called a reflection about (or over, or through) the x -axis.

Given a function f ( x ) , f ( x ) , a new function g ( x ) = f ( − x ) g ( x ) = f ( − x ) is a horizontal reflection of the function f ( x ) , f ( x ) , sometimes called a reflection about the y -axis.

Given a function, reflect the graph both vertically and horizontally.

  • Multiply all outputs by –1 for a vertical reflection. The new graph is a reflection of the original graph about the x -axis.
  • Multiply all inputs by –1 for a horizontal reflection. The new graph is a reflection of the original graph about the y -axis.

Reflecting a Graph Horizontally and Vertically

Reflect the graph of s ( t ) = t s ( t ) = t (a) vertically and (b) horizontally.

Reflecting the graph vertically means that each output value will be reflected over the horizontal t- axis as shown in Figure 13 .

Because each output value is the opposite of the original output value, we can write

Notice that this is an outside change, or vertical shift, that affects the output s ( t ) s ( t ) values, so the negative sign belongs outside of the function.

Reflecting horizontally means that each input value will be reflected over the vertical axis as shown in Figure 14 .

Because each input value is the opposite of the original input value, we can write

Notice that this is an inside change or horizontal change that affects the input values, so the negative sign is on the inside of the function.

Note that these transformations can affect the domain and range of the functions. While the original square root function has domain [ 0 , ∞ ) [ 0 , ∞ ) and range [ 0 , ∞ ) , [ 0 , ∞ ) , the vertical reflection gives the V ( t ) V ( t ) function the range ( − ∞ , 0 ] ( − ∞ , 0 ] and the horizontal reflection gives the H ( t ) H ( t ) function the domain ( − ∞ , 0 ] . ( − ∞ , 0 ] .

Reflect the graph of f ( x ) = | x − 1 | f ( x ) = | x − 1 | (a) vertically and (b) horizontally.

Reflecting a Tabular Function Horizontally and Vertically

A function f ( x ) f ( x ) is given as Table 6 . Create a table for the functions below.

  • ⓐ g ( x ) = − f ( x ) g ( x ) = − f ( x )
  • ⓑ h ( x ) = f ( − x ) h ( x ) = f ( − x )

For g ( x ) , g ( x ) , the negative sign outside the function indicates a vertical reflection, so the x -values stay the same and each output value will be the opposite of the original output value. See Table 7 .

For h ( x ) , h ( x ) , the negative sign inside the function indicates a horizontal reflection, so each input value will be the opposite of the original input value and the h ( x ) h ( x ) values stay the same as the f ( x ) f ( x ) values. See Table 8 .

A function f ( x ) f ( x ) is given as Table 9 . Create a table for the functions below.

Applying a Learning Model Equation

A common model for learning has an equation similar to k ( t ) = − 2 − t + 1 , k ( t ) = − 2 − t + 1 , where k k is the percentage of mastery that can be achieved after t t practice sessions. This is a transformation of the function f ( t ) = 2 t f ( t ) = 2 t shown in Figure 15 . Sketch a graph of k ( t ) . k ( t ) .

This equation combines three transformations into one equation.

  • A horizontal reflection: f ( − t ) = 2 − t f ( − t ) = 2 − t
  • A vertical reflection: − f ( − t ) = − 2 − t − f ( − t ) = − 2 − t
  • A vertical shift: − f ( − t ) + 1 = − 2 − t + 1 − f ( − t ) + 1 = − 2 − t + 1

We can sketch a graph by applying these transformations one at a time to the original function. Let us follow two points through each of the three transformations. We will choose the points (0, 1) and (1, 2).

  • First, we apply a horizontal reflection: (0, 1) (–1, 2).
  • Then, we apply a vertical reflection: (0, −1) (-1, –2).
  • Finally, we apply a vertical shift: (0, 0) (-1, -1).

This means that the original points, (0,1) and (1,2) become (0,0) and (-1,-1) after we apply the transformations.

In Figure 16 , the first graph results from a horizontal reflection. The second results from a vertical reflection. The third results from a vertical shift up 1 unit.

As a model for learning, this function would be limited to a domain of t ≥ 0 , t ≥ 0 , with corresponding range [ 0 , 1 ) . [ 0 , 1 ) .

Given the toolkit function f ( x ) = x 2 , f ( x ) = x 2 , graph g ( x ) = − f ( x ) g ( x ) = − f ( x ) and h ( x ) = f ( − x ) . h ( x ) = f ( − x ) . Take note of any surprising behavior for these functions.

Determining Even and Odd Functions

Some functions exhibit symmetry so that reflections result in the original graph. For example, horizontally reflecting the toolkit functions f ( x ) = x 2 f ( x ) = x 2 or f ( x ) = | x | f ( x ) = | x | will result in the original graph. We say that these types of graphs are symmetric about the y -axis. Functions whose graphs are symmetric about the y -axis are called even functions.

If the graphs of f ( x ) = x 3 f ( x ) = x 3 or f ( x ) = 1 x f ( x ) = 1 x were reflected over both axes, the result would be the original graph, as shown in Figure 17 .

We say that these graphs are symmetric about the origin. A function with a graph that is symmetric about the origin is called an odd function .

Note: A function can be neither even nor odd if it does not exhibit either symmetry. For example, f ( x ) = 2 x f ( x ) = 2 x is neither even nor odd. Also, the only function that is both even and odd is the constant function f ( x ) = 0. f ( x ) = 0.

Even and Odd Functions

A function is called an even function if for every input x x

The graph of an even function is symmetric about the y - y - axis.

A function is called an odd function if for every input x x

The graph of an odd function is symmetric about the origin.

Given the formula for a function, determine if the function is even, odd, or neither.

  • Determine whether the function satisfies f ( x ) = f ( − x ) . f ( x ) = f ( − x ) . If it does, it is even.
  • Determine whether the function satisfies f ( x ) = − f ( − x ) . f ( x ) = − f ( − x ) . If it does, it is odd.
  • If the function does not satisfy either rule, it is neither even nor odd.

Determining whether a Function Is Even, Odd, or Neither

Is the function f ( x ) = x 3 + 2 x f ( x ) = x 3 + 2 x even, odd, or neither?

Without looking at a graph, we can determine whether the function is even or odd by finding formulas for the reflections and determining if they return us to the original function. Let’s begin with the rule for even functions.

This does not return us to the original function, so this function is not even. We can now test the rule for odd functions.

Because − f ( − x ) = f ( x ) , − f ( − x ) = f ( x ) , this is an odd function.

Consider the graph of f f in Figure 18 . Notice that the graph is symmetric about the origin. For every point ( x , y ) ( x , y ) on the graph, the corresponding point ( − x , − y ) ( − x , − y ) is also on the graph. For example, (1, 3) is on the graph of f , f , and the corresponding point ( −1 , −3 ) ( −1 , −3 ) is also on the graph.

Is the function f ( s ) = s 4 + 3 s 2 + 7 f ( s ) = s 4 + 3 s 2 + 7 even, odd, or neither?

Graphing Functions Using Stretches and Compressions

Adding a constant to the inputs or outputs of a function changed the position of a graph with respect to the axes, but it did not affect the shape of a graph. We now explore the effects of multiplying the inputs or outputs by some quantity.

We can transform the inside (input values) of a function or we can transform the outside (output values) of a function. Each change has a specific effect that can be seen graphically.

Vertical Stretches and Compressions

When we multiply a function by a positive constant, we get a function whose graph is stretched or compressed vertically in relation to the graph of the original function. If the constant is greater than 1, we get a vertical stretch ; if the constant is between 0 and 1, we get a vertical compression . Figure 19 shows a function multiplied by constant factors 2 and 0.5 and the resulting vertical stretch and compression.

Given a function f ( x ) , f ( x ) , a new function g ( x ) = a f ( x ) , g ( x ) = a f ( x ) , where a a is a constant, is a vertical stretch or vertical compression of the function f ( x ) . f ( x ) .

  • If a > 1 , a > 1 , then the graph will be stretched.
  • If 0 < a < 1 , 0 < a < 1 , then the graph will be compressed.
  • If a < 0 , a < 0 , then there will be combination of a vertical stretch or compression with a vertical reflection.

Given a function, graph its vertical stretch.

  • Identify the value of a . a .
  • Multiply all range values by a . a .

If a > 1 , a > 1 , the graph is stretched by a factor of a . a .

If 0 < a < 1 , 0 < a < 1 , the graph is compressed by a factor of a . a .

If a < 0 , a < 0 , the graph is either stretched or compressed and also reflected about the x -axis.

Graphing a Vertical Stretch

A function P ( t ) P ( t ) models the population of fruit flies. The graph is shown in Figure 20 .

A scientist is comparing this population to another population, Q , Q , whose growth follows the same pattern, but is twice as large. Sketch a graph of this population.

Because the population is always twice as large, the new population’s output values are always twice the original function’s output values. Graphically, this is shown in Figure 21 .

If we choose four reference points, (0, 1), (3, 3), (6, 2) and (7, 0) we will multiply all of the outputs by 2.

The following shows where the new points for the new graph will be located.

Symbolically, the relationship is written as

This means that for any input t , t , the value of the function Q Q is twice the value of the function P . P . Notice that the effect on the graph is a vertical stretching of the graph, where every point doubles its distance from the horizontal axis. The input values, t , t , stay the same while the output values are twice as large as before.

Given a tabular function and assuming that the transformation is a vertical stretch or compression, create a table for a vertical compression.

  • Determine the value of a . a .
  • Multiply all of the output values by a . a .

Finding a Vertical Compression of a Tabular Function

A function f f is given as Table 10 . Create a table for the function g ( x ) = 1 2 f ( x ) . g ( x ) = 1 2 f ( x ) .

The formula g ( x ) = 1 2 f ( x ) g ( x ) = 1 2 f ( x ) tells us that the output values of g g are half of the output values of f f with the same inputs. For example, we know that f ( 4 ) = 3. f ( 4 ) = 3. Then

We do the same for the other values to produce Table 11 .

The result is that the function g ( x ) g ( x ) has been compressed vertically by 1 2 . 1 2 . Each output value is divided in half, so the graph is half the original height.

A function f f is given as Table 12 . Create a table for the function g ( x ) = 3 4 f ( x ) . g ( x ) = 3 4 f ( x ) .

Recognizing a Vertical Stretch

The graph in Figure 22 is a transformation of the toolkit function f ( x ) = x 3 . f ( x ) = x 3 . Relate this new function g ( x ) g ( x ) to f ( x ) , f ( x ) , and then find a formula for g ( x ) . g ( x ) .

When trying to determine a vertical stretch or shift, it is helpful to look for a point on the graph that is relatively clear. In this graph, it appears that g ( 2 ) = 2. g ( 2 ) = 2. With the basic cubic function at the same input, f ( 2 ) = 2 3 = 8. f ( 2 ) = 2 3 = 8. Based on that, it appears that the outputs of g g are 1 4 1 4 the outputs of the function f f because g ( 2 ) = 1 4 f ( 2 ) . g ( 2 ) = 1 4 f ( 2 ) . From this we can fairly safely conclude that g ( x ) = 1 4 f ( x ) . g ( x ) = 1 4 f ( x ) .

We can write a formula for g g by using the definition of the function f . f .

Write the formula for the function that we get when we stretch the identity toolkit function by a factor of 3, and then shift it down by 2 units.

Horizontal Stretches and Compressions

Now we consider changes to the inside of a function. When we multiply a function’s input by a positive constant, we get a function whose graph is stretched or compressed horizontally in relation to the graph of the original function. If the constant is between 0 and 1, we get a horizontal stretch ; if the constant is greater than 1, we get a horizontal compression of the function.

Given a function y = f ( x ) , y = f ( x ) , the form y = f ( b x ) y = f ( b x ) results in a horizontal stretch or compression. Consider the function y = x 2 . y = x 2 . Observe Figure 23 . The graph of y = ( 0.5 x ) 2 y = ( 0.5 x ) 2 is a horizontal stretch of the graph of the function y = x 2 y = x 2 by a factor of 2. The graph of y = ( 2 x ) 2 y = ( 2 x ) 2 is a horizontal compression of the graph of the function y = x 2 y = x 2 by a factor of 1 2 1 2 .

Given a function f ( x ) , f ( x ) , a new function g ( x ) = f ( b x ) , g ( x ) = f ( b x ) , where b b is a constant, is a horizontal stretch or horizontal compression of the function f ( x ) . f ( x ) .

  • If b > 1 , b > 1 , then the graph will be compressed by 1 b . 1 b .
  • If 0 < b < 1 , 0 < b < 1 , then the graph will be stretched by 1 b . 1 b .
  • If b < 0 , b < 0 , then there will be combination of a horizontal stretch or compression with a horizontal reflection.

Given a description of a function, sketch a horizontal compression or stretch.

  • Write a formula to represent the function.
  • Set g ( x ) = f ( b x ) g ( x ) = f ( b x ) where b > 1 b > 1 for a compression or 0 < b < 1 0 < b < 1 for a stretch.

Graphing a Horizontal Compression

Suppose a scientist is comparing a population of fruit flies to a population that progresses through its lifespan twice as fast as the original population. In other words, this new population, R , R , will progress in 1 hour the same amount as the original population does in 2 hours, and in 2 hours, it will progress as much as the original population does in 4 hours. Sketch a graph of this population.

Symbolically, we could write

See Figure 24 for a graphical comparison of the original population and the compressed population.

Note that the effect on the graph is a horizontal compression where all input values are half of their original distance from the vertical axis.

Finding a Horizontal Stretch for a Tabular Function

A function f ( x ) f ( x ) is given as Table 13 . Create a table for the function g ( x ) = f ( 1 2 x ) . g ( x ) = f ( 1 2 x ) .

The formula g ( x ) = f ( 1 2 x ) g ( x ) = f ( 1 2 x ) tells us that the output values for g g are the same as the output values for the function f f at an input half the size. Notice that we do not have enough information to determine g ( 2 ) g ( 2 ) because g ( 2 ) = f ( 1 2 ⋅ 2 ) = f ( 1 ) , g ( 2 ) = f ( 1 2 ⋅ 2 ) = f ( 1 ) , and we do not have a value for f ( 1 ) f ( 1 ) in our table. Our input values to g g will need to be twice as large to get inputs for f f that we can evaluate. For example, we can determine g ( 4 ) . g ( 4 ) .

We do the same for the other values to produce Table 14 .

Figure 25 shows the graphs of both of these sets of points.

Because each input value has been doubled, the result is that the function g ( x ) g ( x ) has been stretched horizontally by a factor of 2.

Recognizing a Horizontal Compression on a Graph

Relate the function g ( x ) g ( x ) to f ( x ) f ( x ) in Figure 26 .

The graph of g ( x ) g ( x ) looks like the graph of f ( x ) f ( x ) horizontally compressed. Because f ( x ) f ( x ) ends at ( 6 , 4 ) ( 6 , 4 ) and g ( x ) g ( x ) ends at ( 2 , 4 ) , ( 2 , 4 ) , we can see that the x - x - values have been compressed by 1 3 , 1 3 , because 6 ( 1 3 ) = 2. 6 ( 1 3 ) = 2. We might also notice that g ( 2 ) = f ( 6 ) g ( 2 ) = f ( 6 ) and g ( 1 ) = f ( 3 ) . g ( 1 ) = f ( 3 ) . Either way, we can describe this relationship as g ( x ) = f ( 3 x ) . g ( x ) = f ( 3 x ) . This is a horizontal compression by 1 3 . 1 3 .

Notice that the coefficient needed for a horizontal stretch or compression is the reciprocal of the stretch or compression. So to stretch the graph horizontally by a scale factor of 4, we need a coefficient of 1 4 1 4 in our function: f ( 1 4 x ) . f ( 1 4 x ) . This means that the input values must be four times larger to produce the same result, requiring the input to be larger, causing the horizontal stretching.

Write a formula for the toolkit square root function horizontally stretched by a factor of 3.

Performing a Sequence of Transformations

When combining transformations, it is very important to consider the order of the transformations. For example, vertically shifting by 3 and then vertically stretching by 2 does not create the same graph as vertically stretching by 2 and then vertically shifting by 3, because when we shift first, both the original function and the shift get stretched, while only the original function gets stretched when we stretch first.

When we see an expression such as 2 f ( x ) + 3 , 2 f ( x ) + 3 , which transformation should we start with? The answer here follows nicely from the order of operations. Given the output value of f ( x ) , f ( x ) , we first multiply by 2, causing the vertical stretch, and then add 3, causing the vertical shift. In other words, multiplication before addition.

Horizontal transformations are a little trickier to think about. When we write g ( x ) = f ( 2 x + 3 ) , g ( x ) = f ( 2 x + 3 ) , for example, we have to think about how the inputs to the function g g relate to the inputs to the function f . f . Suppose we know f ( 7 ) = 12. f ( 7 ) = 12. What input to g g would produce that output? In other words, what value of x x will allow g ( x ) = f ( 2 x + 3 ) = 12 ? g ( x ) = f ( 2 x + 3 ) = 12 ? We would need 2 x + 3 = 7. 2 x + 3 = 7. To solve for x , x , we would first subtract 3, resulting in a horizontal shift, and then divide by 2, causing a horizontal compression.

This format ends up being very difficult to work with, because it is usually much easier to horizontally stretch a graph before shifting. We can work around this by factoring inside the function.

Let’s work through an example.

We can factor out a 2.

Now we can more clearly observe a horizontal shift to the left 2 units and a horizontal compression. Factoring in this way allows us to horizontally stretch first and then shift horizontally.

Combining Transformations

When combining vertical transformations written in the form a f ( x ) + k , a f ( x ) + k , first vertically stretch by a a and then vertically shift by k . k .

When combining horizontal transformations written in the form f ( b x - h ) , f ( b x - h ) , first horizontally shift by h b h b and then horizontally stretch by 1 b . 1 b .

When combining horizontal transformations written in the form f ( b ( x - h ) ) , f ( b ( x - h ) ) , first horizontally stretch by 1 b 1 b and then horizontally shift by h . h .

Horizontal and vertical transformations are independent. It does not matter whether horizontal or vertical transformations are performed first.

Finding a Triple Transformation of a Tabular Function

Given Table 15 for the function f ( x ) , f ( x ) , create a table of values for the function g ( x ) = 2 f ( 3 x ) + 1. g ( x ) = 2 f ( 3 x ) + 1.

There are three steps to this transformation, and we will work from the inside out. Starting with the horizontal transformations, f ( 3 x ) f ( 3 x ) is a horizontal compression by 1 3 , 1 3 , which means we multiply each x - x - value by 1 3 . 1 3 . See Table 16 .

Looking now to the vertical transformations, we start with the vertical stretch, which will multiply the output values by 2. We apply this to the previous transformation. See Table 17 .

Finally, we can apply the vertical shift, which will add 1 to all the output values. See Table 18 .

Finding a Triple Transformation of a Graph

Use the graph of f ( x ) f ( x ) in Figure 27 to sketch a graph of k ( x ) = f ( 1 2 x + 1 ) − 3. k ( x ) = f ( 1 2 x + 1 ) − 3.

To simplify, let’s start by factoring out the inside of the function.

By factoring the inside, we can first horizontally stretch by 2, as indicated by the 1 2 1 2 on the inside of the function. Remember that twice the size of 0 is still 0, so the point (0,2) remains at (0,2) while the point (2,0) will stretch to (4,0). See Figure 28 .

Next, we horizontally shift left by 2 units, as indicated by x + 2. x + 2. See Figure 29 .

Last, we vertically shift down by 3 to complete our sketch, as indicated by the − 3 − 3 on the outside of the function. See Figure 30 .

Access this online resource for additional instruction and practice with transformation of functions.

  • Function Transformations

1.5 Section Exercises

When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal shift from a vertical shift?

When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal stretch from a vertical stretch?

When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal compression from a vertical compression?

When examining the formula of a function that is the result of multiple transformations, how can you tell a reflection with respect to the x -axis from a reflection with respect to the y -axis?

How can you determine whether a function is odd or even from the formula of the function?

Write a formula for the function obtained when the graph of f ( x ) = x f ( x ) = x is shifted up 1 unit and to the left 2 units.

Write a formula for the function obtained when the graph of f ( x ) = | x | f ( x ) = | x | is shifted down 3 units and to the right 1 unit.

Write a formula for the function obtained when the graph of f ( x ) = 1 x f ( x ) = 1 x is shifted down 4 units and to the right 3 units.

Write a formula for the function obtained when the graph of f ( x ) = 1 x 2 f ( x ) = 1 x 2 is shifted up 2 units and to the left 4 units.

For the following exercises, describe how the graph of the function is a transformation of the graph of the original function f . f .

y = f ( x − 49 ) y = f ( x − 49 )

y = f ( x + 43 ) y = f ( x + 43 )

y = f ( x + 3 ) y = f ( x + 3 )

y = f ( x − 4 ) y = f ( x − 4 )

y = f ( x ) + 5 y = f ( x ) + 5

y = f ( x ) + 8 y = f ( x ) + 8

y = f ( x ) − 2 y = f ( x ) − 2

y = f ( x ) − 7 y = f ( x ) − 7

y = f ( x − 2 ) + 3 y = f ( x − 2 ) + 3

y = f ( x + 4 ) − 1 y = f ( x + 4 ) − 1

For the following exercises, determine the interval(s) on which the function is increasing and decreasing.

f ( x ) = 4 ( x + 1 ) 2 − 5 f ( x ) = 4 ( x + 1 ) 2 − 5

g ( x ) = 5 ( x + 3 ) 2 − 2 g ( x ) = 5 ( x + 3 ) 2 − 2

a ( x ) = − x + 4 a ( x ) = − x + 4

k ( x ) = − 3 x − 1 k ( x ) = − 3 x − 1

For the following exercises, use the graph of f ( x ) = 2 x f ( x ) = 2 x shown in Figure 31 to sketch a graph of each transformation of f ( x ) . f ( x ) .

g ( x ) = 2 x + 1 g ( x ) = 2 x + 1

h ( x ) = 2 x − 3 h ( x ) = 2 x − 3

w ( x ) = 2 x − 1 w ( x ) = 2 x − 1

For the following exercises, sketch a graph of the function as a transformation of the graph of one of the toolkit functions.

f ( t ) = ( t + 1 ) 2 − 3 f ( t ) = ( t + 1 ) 2 − 3

h ( x ) = | x − 1 | + 4 h ( x ) = | x − 1 | + 4

k ( x ) = ( x − 2 ) 3 − 1 k ( x ) = ( x − 2 ) 3 − 1

m ( t ) = 3 + t + 2 m ( t ) = 3 + t + 2

Tabular representations for the functions f , g , f , g , and h h are given below. Write g ( x ) g ( x ) and h ( x ) h ( x ) as transformations of f ( x ) . f ( x ) .

For the following exercises, write an equation for each graphed function by using transformations of the graphs of one of the toolkit functions.

For the following exercises, use the graphs of transformations of the square root function to find a formula for each of the functions.

For the following exercises, use the graphs of the transformed toolkit functions to write a formula for each of the resulting functions.

For the following exercises, determine whether the function is odd, even, or neither.

f ( x ) = 3 x 4 f ( x ) = 3 x 4

g ( x ) = x g ( x ) = x

h ( x ) = 1 x + 3 x h ( x ) = 1 x + 3 x

f ( x ) = ( x − 2 ) 2 f ( x ) = ( x − 2 ) 2

g ( x ) = 2 x 4 g ( x ) = 2 x 4

h ( x ) = 2 x − x 3 h ( x ) = 2 x − x 3

For the following exercises, describe how the graph of each function is a transformation of the graph of the original function f . f .

g ( x ) = − f ( x ) g ( x ) = − f ( x )

g ( x ) = f ( − x ) g ( x ) = f ( − x )

g ( x ) = 4 f ( x ) g ( x ) = 4 f ( x )

g ( x ) = 6 f ( x ) g ( x ) = 6 f ( x )

g ( x ) = f ( 5 x ) g ( x ) = f ( 5 x )

g ( x ) = f ( 2 x ) g ( x ) = f ( 2 x )

g ( x ) = f ( 1 3 x ) g ( x ) = f ( 1 3 x )

g ( x ) = f ( 1 5 x ) g ( x ) = f ( 1 5 x )

g ( x ) = 3 f ( − x ) g ( x ) = 3 f ( − x )

g ( x ) = − f ( 3 x ) g ( x ) = − f ( 3 x )

For the following exercises, write a formula for the function g g that results when the graph of a given toolkit function is transformed as described.

The graph of f ( x ) = | x | f ( x ) = | x | is reflected over the y y - axis and horizontally compressed by a factor of 1 4 1 4 .

The graph of f ( x ) = x f ( x ) = x is reflected over the x x -axis and horizontally stretched by a factor of 2.

The graph of f ( x ) = 1 x 2 f ( x ) = 1 x 2 is vertically compressed by a factor of 1 3 , 1 3 , then shifted to the left 2 units and down 3 units.

The graph of f ( x ) = 1 x f ( x ) = 1 x is vertically stretched by a factor of 8, then shifted to the right 4 units and up 2 units.

The graph of f ( x ) = x 2 f ( x ) = x 2 is vertically compressed by a factor of 1 2 , 1 2 , then shifted to the right 5 units and up 1 unit.

The graph of f ( x ) = x 2 f ( x ) = x 2 is horizontally stretched by a factor of 3, then shifted to the left 4 units and down 3 units.

For the following exercises, describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation.

g ( x ) = 4 ( x + 1 ) 2 − 5 g ( x ) = 4 ( x + 1 ) 2 − 5

h ( x ) = − 2 | x − 4 | + 3 h ( x ) = − 2 | x − 4 | + 3

m ( x ) = 1 2 x 3 m ( x ) = 1 2 x 3

n ( x ) = 1 3 | x − 2 | n ( x ) = 1 3 | x − 2 |

p ( x ) = ( 1 3 x ) 3 − 3 p ( x ) = ( 1 3 x ) 3 − 3

q ( x ) = ( 1 4 x ) 3 + 1 q ( x ) = ( 1 4 x ) 3 + 1

For the following exercises, use the graph in Figure 32 to sketch the given transformations.

g ( x ) = f ( x ) − 2 g ( x ) = f ( x ) − 2

g ( x ) = f ( x + 1 ) g ( x ) = f ( x + 1 )

g ( x ) = f ( x − 2 ) g ( x ) = f ( x − 2 )

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  • Authors: Jay Abramson
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  • Book title: Precalculus 2e
  • Publication date: Dec 21, 2021
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  • Book URL: https://openstax.org/books/precalculus-2e/pages/1-introduction-to-functions
  • Section URL: https://openstax.org/books/precalculus-2e/pages/1-5-transformation-of-functions

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1.12 Transformations of Functions

1 min read • august 7, 2023

Transformations! Now, let’s get familiar with additive and multiplicative transformations, which encapsulate translations, dilations, and reflections.

Additive Transformations (Translations)

An additive transformation of a function is a transformation that involves adding or subtracting a constant value to the function.

1️⃣ Vertical Translations

The function g ( x ) = f ( x ) + k g(x) = f(x) + k g ( x ) = f ( x ) + k represents an additive transformation of the function f. In this case, the function f is being shifted vertically by k units. The value of k determines the magnitude and direction of the shift.

The result of this additive transformation is a vertical translation of the graph of f. A vertical translation is a transformation that involves moving the graph of a function up or down along the y-axis. In this case, the graph of f is being moved up or down by k units. ↕️

To visualize the effect of the additive transformation, we can plot the graphs of f and g on the same set of axes. The graph of f would appear in its original position, while the graph of g would be shifted horizontally by k units and vertically by k units.

q-28-2-1634736085.png

Image Courtesy of Cuemath

2️⃣ Horizontal Translations

The function g ( x ) = f ( x + h ) g(x) = f(x + h) g ( x ) = f ( x + h ) also represents an additive transformation of the function f.

This time, the result of this additive transformation is a horizontal translation of the graph of f. A horizontal translation is a transformation that involves moving the graph of a function left or right along the x-axis. In this case, the graph of f is being moved to the left or right by h units, depending on the sign of h. ↔️

To visualize the effect of the additive transformation, we can plot the graphs of f and g on the same set of axes. The graph of f would appear in its original position, while the graph of g would be shifted horizontally by h units and vertically by 0 units.

f8f9b2eefc60479b807b08743c747594.jpeg

Image Courtesy of Quora

For both cases, it’s crucial to note that the shape of the graph of f remains unchanged by the additive transformation. Only its position on the coordinate plane is altered. 🚨 Therefore, if we know the graph of f, we can easily sketch the graph of g by applying the appropriate horizontal and vertical shifts.

🪞 Multiplicative Transformations (Dilations and Reflections)

A multiplicative transformation involves multiplying the function by a constant value.

1️⃣ Vertical Dilations

The function g ( x ) = a f ( x ) g(x) = af(x) g ( x ) = a f ( x ) , where a is a non-zero constant, represents a multiplicative transformation of the function f. In this case, the function f is being scaled vertically by a factor of |a|, which means the distance between the function and the x-axis is increased or decreased by a factor of |a|.

The result of this multiplicative transformation is a vertical dilation of the graph of f, a transformation that involves stretching or shrinking the graph of a function vertically.

  • If a > 1 a > 1 a > 1 , the dilation causes the graph of f to be stretched vertically, making the curve appear "taller." 🔼
  • If 0 < a < 1 0 < a < 1 0 < a < 1 , the dilation causes the graph of f to be shrunk vertically, making the curve appear "shorter.” 🔽
  • If a < 0 a < 0 a < 0 , the transformation also involves a reflection over the x-axis, which means the graph of f is flipped over the x-axis. 🔁

To visualize the effect of the multiplicative transformation, we can plot the graphs of f and g on the same set of axes. The graph of f would appear in its original position, while the graph of g would be scaled vertically by a factor of |a|.

205680db2491940ca84a05d3de631554.png

Image Courtesy of Github

2️⃣ Horizontal Dilations

The function g ( x ) = f ( b x ) g(x) = f(bx) g ( x ) = f ( b x ) , where b is a non-zero constant, also represents a multiplicative transformation of the function f.

In this case, the function f is being scaled horizontally by a factor of ∣ 1 / b ∣ |1/b| ∣1/ b ∣ , which means the distance between the function and the y-axis is increased or decreased by a factor of ∣ 1 / b ∣ |1/b| ∣1/ b ∣ .

The result of this multiplicative transformation is a horizontal dilation of the graph of f, a transformation that involves stretching or shrinking the graph of a function horizontally .

  • If ∣ b ∣ > 1 |b| > 1 ∣ b ∣ > 1 , the dilation causes the graph of f to be shrunk horizontally, making the curve appear "wider." ▶️
  • If 0 < ∣ b ∣ < 1 0 < |b| < 1 0 < ∣ b ∣ < 1 , the dilation causes the graph of f to be stretched horizontally, making the curve appear "narrower." ◀️
  • If b < 0 b < 0 b < 0 , the transformation also involves a reflection over the y-axis, which means the graph of f is flipped over the y-axis. 🔁

To visualize the effect of the multiplicative transformation, we can plot the graphs of f and g on the same set of axes. The graph of f would appear in its original position, while the graph of g would be scaled horizontally by a factor of ∣ 1 / b ∣ |1/b| ∣1/ b ∣ .

11.jpg

Image Courtesy of Nagwa

For both cases, note that the shape of the graph of f remains unchanged by the multiplicative transformation. Only its size is altered. Therefore, if we know the graph of f, we can easily sketch the graph of g by applying the appropriate vertical dilation and reflection, if necessary.

Working with Transformations

Additive and multiplicative transformations can be combined to create more complex transformations of a function. When these transformations are combined, they result in a combination of horizontal and vertical translations and dilations. This means that the graph of the transformed function is shifted both horizontally and vertically and is also scaled horizontally and/or vertically. 😳

For example, consider the function g ( x ) = a ∗ f ( b x + h ) + k g(x) = a*f(bx + h) + k g ( x ) = a ∗ f ( b x + h ) + k , where a, b, h, and k are constants. This function is a combination of an additive and a multiplicative transformation of the function f. The function f is transformed horizontally by a factor of |1/b| and then horizontally translated by h units. It is also vertically scaled by a factor of a and then vertically translated by k units.

It is important to note that when a function is transformed, its domain and range may change. 😮 The domain of the transformed function may be restricted due to the nature of the transformations.

For example, if a function is reflected over the x-axis, its domain changes from all real numbers to all real numbers except zero. The range of the transformed function may also change due to the vertical scaling.

CNX_Precalc_Figure_02_02_0052.jpg

Image Courtesy of Lumen Learning

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The topics covered are: Lines and Planes, Functions and Transformation of Graphs, Polynomials, Rational Functions, Limits of a Function, Complex Numbers, Exponential Functions, Logarithmic Functions, Conic Sections, Matrices, Sequences and Series, Probability and Combinatorics, Advanced Trigonometry, Vectors and Parametric Equations, and Polar Coordinates.

Parallel Lines, Skew Lines & Planes Proving Parallel Lines

Difference Quotient of Function I Difference Quotient of Function II Domain of a Function Greatest Integer Function I Greatest Integer Function II Piece-Wise Functions I Piece-Wise Functions II Piece-Wise Functions III Even & Odd Functions I Even & Odd Functions II Composite Functions I Composite Functions II Composite Functions III One-to-one Functions Inverse of a Function I Inverse of a Function II Finding the Inverse of a Function or Showing One Does not Exist

Parent Functions & their Graphs Transformation of Linear Functions Transformation of Quadratic Functions Horizontal & Vertical Graph Transformations Horizontal & Vertical Graph Stretches & Compressions Reflective Transformations Examples of Graph Transformations

Power Functions Polynomial Functions Graphing Polynomial Functions I Graphing Polynomial Functions II Families of Polynomials Graphing Polynomials with Repeated Factors Finding the Equation of a Polynomial Function Finding Maximum & Minimum Values of Polynomial Functions Find the Zeros of a Polynomial Function I Find the Zeros of a Polynomial Function II Conjugate Zeros Theorem Reciprocal of a Function

Introduction to Rational Functions Limits of Rational Functions Graphing Rational Functions I Graphing Rational Functions II Horizontal Asymptotes I Horizontal Asymptotes II Vertical Asymptotes Vertical & Horizontal Asymptotes Oblique or Slant Asymptotes Find all Asymptotes of a Rational Function Graphing Rational Functions with Holes Rational Function Problems

Introduction to Limits of Functions Limits of Rational Functions Examples to Learn about Limits Calculating Limits using Different Techniques I Calculating Limits using Different Techniques II Limit Laws to Evaluate a Limit One Sided Limits Squeeze Theorem for Limits The Limit definition of Continuity

Introduction to Complex Numbers I Introduction to Complex Numbers II Adding & Subtracting Complex Numbers Multiplying Complex Numbers Dividing Complex Numbers Trigonometric or Polar Form of Complex Numbers Converting Complex Numbers between Trigonometric Form & Rectangular Multiplying & Dividing Complex Numbers in Trigonometric or Polar Form DeMoivre’s Theorem & Euler Formula Roots of a Complex Number Complex Quadratic Equations I Complex Quadratic Equations II Simplify Complex Rational Expressions

Introduction to Exponential Functions Graphing Exponential Functions I Graphing Exponential Functions II Applying Exponential Functions I Applying Exponential Functions II Solving Exponential Equations with the Same Base Solving Exponential Equations with Different Bases I Solving Exponential Equations with Different Bases II Solving Exponential Equations with Different Bases III

Introduction to Logarithmic Functions Exponential & Logarithmic Functions Logarithm Review Product Rule Logarithm Review Product & Quotient Rules Logarithm Review Power Rule Logarithm Review Change of Base Rule Properties of Logarithms (or Rules of Logarithms) Proof of Logarithm Properties Expanding Logarithmic Expressions Simplifying (or Condensing) Logarithmic Expressions Simplifying & Expanding Logarithmic Expressions Solving Logarithmic Equations I Solving Logarithmic Equations II Graphs of Logarithmic Functions

Introduction to Conic Sections Conic Sections - Circles I Conic Sections - Circles II Conic Sections - Ellipses I Conic Sections - Ellipses II Conic Sections - Parabolas Conic Sections - Hyperbolas I Conic Sections - Hyperbolas II Conic Sections - Hyperbolas III Conic Sections Summary Identify & Graph Conic Sections

Introduction to Matrices Matrix Addition & Subtraction I Matrix Addition & Subtraction II Matrix Scalar Multiplication Matrix Multiplication I Matrix Multiplication II Matrix Multiplication III Identity Matrix Determinant of a 2x2 Matrix Determinant of a 3x3 Matrix I Determinant of a 3x3 Matrix II Simplifying Determinant Inverse of 2x2 Matrix Inverse of 3x3 Matrix Singular Matrix - A Matrix with no Inverse Solving a 2x2 System of Equations Using a Matrix Inverse I Solving a 2x2 System of Equations Using a Matrix Inverse II Solving a 3x3 System of Equations Using a Matrix Inverse Using Gauss-Jordan to Solve a System of Three Linear Equations Row Reducing a Matrix to solve a System of Equations Solving a System of Equations using Matrix Row Transformations Cramer’s Rule Using Determinant to find the Area of a Parallelogram Using Determinant to find the Area of a Triangle & a Polygon Cross Product of Vectors & Determinants - area of triangles and parallelograms

Introduction to Sequences Arithmetic Sequences I Arithmetic Sequences II Arithmetic Sequences Finding the nth Term Geometric Sequences I Geometric Sequences II Geometric Sequences Finding the nth Term Recursion Sequences

Series and Summation Arithmetic Series I Arithmetic Series II Geometric Series I Geometric Series II Infinite Geometric Series

Mathematical Induction Examples of Mathematical Induction I Examples of Mathematical Induction II

Introduction to Probability I Introduction to Probability II Fundamental Counting Principles Permutations I Permutations II Permutations III Combinations I Combinations II Permutations & Combinations Probability using Permutations & Combinations Probability of Multiple Events Probability of Independent Events Probability of Dependent Events Probability of Complementary Events Conditional Probability I Conditional Probability II Conditional Probability III Pascal’s Triangle & the Binomial Theorem Binomial Theorem I Binomial Theorem II

Inverse Trigonometric Functions I Inverse Trigonometric Functions II Factoring Trigonometric Equations Trig Identities cofunctions, reciprocal, quotient, ratio, Pythagorean, & even/odd Sum & Difference Identities I Sum & Difference Identities II Sine and Cosine Addition Formulas (Proofs) Double Angle Identities I Double Angle Identities II Power Reducing Identities Half Angle Identities Half Angle & Double Angle Examples

Geometric Representation of Vectors Algebraic Representation of Vectors Components of a Vector Adding Vectors Graphically or Head-to-Tail Method Adding Vectors using Components I Adding Vectors using Components II Vector Addition & Scalar Multiplication Vector Magnitude & Direction The Resultant of Two Forces Components of a Force Solve Navigation Problems using Vectors Unit Vectors I Unit Vectors II Vector Equation of a Line Parametric Equations & Motion Parameterize a Line Segment & a Circle Parametric Equations Dot Product of Vectors Angle between Two Vectors Vectors in Three Dimensions Lines in 3D Coordinate Systems Vectors and Planes I Vectors and Planes II

Introduction to Polar Coordinates Conversion between Polar Coordinates & Rectangular Coordinates Polar Coordinates Distance Formula Equations of Lines in Polar Coordinates Convert between Polar Equations & Rectangular Equations Symmetry of Polar Graphs Graphing Polar Equations Families of Polar Curves: Circles, Cardiods, Limacon, Roses & Conic Sections

IIT JEE Trigonometry Problem IIT JEE Perpendicular Planes IIT JEE Complex Root Probability IIT JEE Position Vectors ITT JEE Matrix Equations IIT JEE Integral Limit IIT JEE Algebraic Manipulation IIT JEE Function Maxima IIT JEE Diameter Slope IIT JEE Hairy Trig & Algebra IIT JEE Complex Numbers IIT JEE Differentiability & Boundedness IIT JEE Integral with Binomial Expansion IIT JEE Symmetric & Skew-Symmetric Matrices IIT JEE Trace & Determinant IIT JEE Divisible Determinants IIT JEE Circle Hyperbola Intersection IIT JEE Circle Hyperbola Common Tangent IIT JEE Trigonometric Constraints IIT JEE Trigonometric Maximum Vector Triple Product Expansion (very optional) IIT JEE Lagrange’s Formula Tangent Line Hyperbola Relationship (very optional) 2010 IIT JEE Paper 1 Problem 50 Hyperbola Eccentricity Normal vector from plane equation Point distance to plane Distance Between Planes Periodic Definite Integral Complex Determinant Example Series Sum Example Trigonometric System Example Simple Differential Equation Example

PreCalculus Calculator with step by step solutions Relations, Functions, Operations on Functions, Points, Lines, & Line Segments, Absolute Value Expressions & Equations, Radical Expressions & Equations, Rational Expressions & Equations, Polynomial & Rational Functions, Factoring Polynomials, Conic Sections, Exponential & Logarithmic Functions, Trigonometry, Analytic Trigonometry, Inequalities, Linear Equations, Systems of Equations, Quadratic Equations Matrices, Sequences & Series, Analytic Geometry in Rectangular Coordinates, Analytic Geometry in Polar Coordinates, Limits & an Introduction to Calculus, Vectors, Number Sets

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Mrs. Snow's Math

Mcneil high school.

  • Parent Letter
  •  Algebra I
  • Math Modeling

Pre Calculus

Chapter 2:  Functions and Their Graphs

Lesson 2.1 and Lesson 2.2 Functions and 2.2 The Graph of a Function

Annotated Notes 2.1 and 2.2

Lesson 2.3 and Lesson 2.4 Properties of Functions and 2.4  Library of Functions; Piecewise-defined Functions

Annotated Notes 2.3 and 2.4

Lesson 2.5  Graphing Techniques:  Transformations

2.5 Annotated In-class Activity

Annotated Notes 2.5

2.5 Worksheet Homework

Chapter 2 Review

Chapter 2 Review Solutions corrected!

Chapter 4:  Polynomial and Rational Functions

Lesson 4.1 Polynomial Functions and Models

Annotated Notes

Lesson 4.2 The Real Zeros of a Polynomial Function

Annotated Notes (includes 4.2 and 4.3) 

Lesson 4.3 Complex Zeros; Fundamental Theorem of Algebra

Lesson 4.4 Properties of Rational Functions

Annotated Notes 4.4 and 4.5  Corrected!

Lesson 4.5 The Graphs of a Rational Function

Lesson 4.6 Polynomial and Rational Inequalities

Chapter 4 Review 2015

Solutions 2015

Lesson 5.1 Composite Functions

Lesson 5.2 One-to-One Functions, Inverse Functions

Annotated Notes 5.1 and 5.2

Chapter 6 Trigonometric Functions

Lesson 6.1 Angles and Their Measure

Lesson 6.2 Trigonometric Functions: Unit Circle Approach

2 Unit Circles   Practice Form

Lesson 6.3 Properties of the Trigonometric Functions

Test 6.1-6.3 Review

6.1-6.3 Review Solutions

Lesson 6.4 Graphs of the Sine and Cosine Functions

Worksheet 6.4  

Lesson 6.5 Graphs of the Tangent, Cotangent, Cosecant and Secant Functions (corrected)

Worksheet 6.5  

Lesson 6.6 Phase Shift

Worksheet 6.6 (revised)  

Test 6.1-6.6 Spiral Review

Solutions Chapter 6 Spiral Test Review

Unit Circle Chart In Order

Lesson 7.1   The Inverse Sine, Cosine, and Tangent Functions

7.1 Extra Examples

Lesson 7.2 The Inverse Trigonometric Function, Continued

Lesson 7.3 Trigonometric Equations

Review 7.1-7.3

Solutions Review 7.1-7.3

Trigonometric Identities Reference Sheet 

Lesson 7.4 Trigonometric Identities

Worksheet Proofs Trig Identities

Lesson 7.5 Sum and Difference Formulas

Annotated Notes  

Lesson 7.6 Double-angle and Half-angle Formulas

Chapter 7 Spiral Review

Solutions Spiral Review

Fall 2015 Final Exam Review

Solutions answers have not been checked for accuracy, if you have a conflict, remember you may be correct!

FINAL EXAM NOTE CARD GUIDELINE

SPRING SEMESTER   

Copyright © 2009. All Rights Reserved.

Home | Class Info | Parent Letter | Algebra I | Algebra II | Pre Calc | Math Modeling | Links | Contact | McNeil HS

pre calc transformations worksheet

Precalculus 9e

pre calc transformations worksheet

Nonrigid transformations

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Precalculus

Unit 1: composite and inverse functions, unit 2: trigonometry, unit 3: complex numbers, unit 4: rational functions, unit 5: conic sections, unit 6: vectors, unit 7: matrices, unit 8: probability and combinatorics, unit 9: series, unit 10: limits and continuity.

IMAGES

  1. Precalculus Transformations Worksheet

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  2. Transformations Math Worksheet

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  3. Transformations Practice Worksheet Precalculus

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  4. Transformations Worksheets with Answers

    pre calc transformations worksheet

  5. Sequence Of Transformations Worksheet Transformations Day 1 Answer Key

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  6. Precalculus Transformations Worksheet

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VIDEO

  1. O LEVELS TRANSFORMATIONS TOPICAL WORKSHEET QUESTIONS 4024/0580

  2. math formula with example

  3. PreCalc Lesson 3.5

  4. Pre Calc A

  5. Pre Calc CP Unit 5 Day 1 Graphing Rational Functions

  6. Combining Transformations Pre Calculus / Advanced Functions 12

COMMENTS

  1. 1-5 Parent Functions and Transformations

    Parent Functions and Transformations Worksheet, Word Docs, & PowerPoints. 1-5 Assignment - Parent Functions and Transformations. 1-5 Bell Work - Parent Functions and Transformations. 1-5 Exit Quiz - Parent Functions and Transformations. 1-5 Guided Notes SE - Parent Functions and Transformations.

  2. Worksheets

    Worksheets. Relations and Functions. Evaluations of Functions. The Horizontal Line Test. Finding the Domain of a Function. Finding the Domain and Range of a Function from its Graph. Piecewise Functions. Average Rate of Change of a Function. Increasing and Decreasing Functions.

  3. PDF Transformations of Graphs Date Period

    Describe the transformations necessary to transform the graph of f(x) (solid line) into that of g(x) (dashed line). 1) x y reflect across the x-axis translate left units 2) x y compress vertically by a factor of translate up units Describe the transformations necessary to transform the graph of f(x) into that of g(x). 3) f (x) x

  4. Mr. Campbell Math

    PRE-CALC 11 PRE-CALC 12. CALC 12. LINKS. More. UNIT 1 TRANSFORMATIONS & FUNCTIONS ... TRANSFORMATIONS & FUNCTIONS. Chapter 1 - Function Transformations. Chapter 2 - Radical Functions. Chapter 3 - Polynomial Functions • UNIT 1 NOTES PACKAGE ... WORKSHEET. ANSWER KEY ...

  5. 4.1 Transformations

    4.1 Transformations. Pre Calc 4.1 Transformations. Watch on. Need a tutor? Click this link and get your first session free!

  6. 1.5 Transformation of Functions

    Combining Vertical and Horizontal Shifts. Now that we have two transformations, we can combine them together. Vertical shifts are outside changes that affect the output ( y-y-) axis values and shift the function up or down.Horizontal shifts are inside changes that affect the input ( x-x-) axis values and shift the function left or right.Combining the two types of shifts will cause the graph of ...

  7. Precalculus

    Chapter 2: Functions and Their Graphs. All assignments are due at the start of the next class period. The first assignments of this school year will be completed on paper: Current assignments are currently not available on Mathxl, the precalculus online homework program. For full credit, use separate paper: please write your full name, class ...

  8. AP Pre-Calculus Study Guide

    1️⃣ Vertical Translations. The function g (x) = f (x) + k g(x) = f (x)+k represents an additive transformation of the function f. In this case, the function f is being shifted vertically by k units. The value of k determines the magnitude and direction of the shift. The result of this additive transformation is a vertical translation of the ...

  9. IXL

    IXL's SmartScore is a dynamic measure of progress towards mastery, rather than a percentage grade. It tracks your skill level as you tackle progressively more difficult questions. Consistently answer questions correctly to reach excellence (90), or conquer the Challenge Zone to achieve mastery (100)! Learn more. 0.

  10. Transformations Precalculus Worksheets & Teaching Resources

    7. $3.50. Zip. This Graphical Transformations Match-Up Activity is designed to help your Pre-Calculus students gain a better understanding of transformations and writing equations. There are 36 task cards in the activity. The cards are sorted into sets which include a graph, an equation, and a description of the.

  11. Free Printable Math Worksheets for Precalculus

    Definition of the derivative. Instantaneous rates of change. Power rule for differentiation. Motion along a line. Approximating area under a curve. Area under a curve by limit of sums. Indefinite integrals. Free Precalculus worksheets created with Infinite Precalculus. Printable in convenient PDF format.

  12. PreCalculus (solutions, examples, worksheets, lessons, videos, activities)

    Videos, examples, solutions, activities and worksheets for studying, practice and review of precalculus, Lines and Planes, Functions and Transformation of Graphs, Polynomials, Rational Functions, Limits of a Function, Complex Numbers, Exponential Functions, Logarithmic Functions, Conic Sections, Matrices, Sequences and Series, Probability and Combinatorics, Advanced Trigonometry, Vectors and ...

  13. PDF Pre-Calculus Section 2.4 Worksheet [Day 2] Name: Sept 2013 Suppose the

    Pre-Calculus Section 2.4 Worksheet [Day 2] Name:_____ Sept 2013 Suppose the graph of is given. Describe the transformations that would be performed on to ... and the indicated transformations are applied to its graph (in the given order). Write the equation for the final transformed graph. 11. ()=; shift left 1, stretch vertically by a factor ...

  14. Fall Notes

    Pre Calculus . Notes are intended to compliment the current text in use at McNeil High School . ... Transformations . 2.5 Annotated In-class Activity. Annotated Notes 2.5. 2.5 Worksheet Homework . ... Worksheet 6.4 . Lesson 6.5 Graphs of the Tangent, Cotangent, Cosecant and Secant Functions ...

  15. Pre Calc Transformations Teaching Resources

    Flamingo Math by Jean Adams. This Graphical Transformations Match-Up Activity is designed to help your Pre - Calculus students gain a better understanding of transformations and writing equations. There are 36 task cards in the activity. The cards are sorted into sets which include a graph, an equation, and a description of the transformed graph.

  16. Infinite Precalculus

    Infinite Precalculus covers all typical Precalculus material and more: trigonometric functions, equations, and identities; parametric equations; polar coordinates; vectors; limits; and more. Over 100 individual topics extend skills from Algebra 2 and introduce Calculus. Test and worksheet generator for Precalculus.

  17. 4.1 Transformations

    4.1 Transformations Notes Key. Application Notes Key. Practice Key. Application Key. Powered by Create your own unique website with customizable templates.

  18. List of Pre Calculus Worksheets

    List of Pre Calculus Worksheets Functions Continuity Extrema, intervals of increase and decrease Power functions Average rates of change Transformations of graphs Piecewise functions Operations Inverses Power, Polynomial, and Rational Functions Graphs, real zeros, and end behavior Dividing polynomial functions The Remainder Theorem and bounds of real zeros

  19. Nonrigid transformations

    If you are in need of technical support, have a question about advertising opportunities, or have a general question, please contact us by phone or submit a message through the form below. Journal. Organizations. AMATYC Review. American Mathematical Association of Two-Year Colleges. American Mathematical Monthly.

  20. 1.12A Translations of Functions

    AP Learning Objectives:1.12.A Construct a function that is an additive and/or multiplicative transformation of another functions. *AP® is a trademark registered and owned by the CollegeBoard, which was not involved in the production of, and does not endorse, this site.

  21. Precalculus

    The Precalculus course covers complex numbers; composite functions; trigonometric functions; vectors; matrices; conic sections; and probability and combinatorics. It also has two optional units on series and limits and continuity. Khan Academy's Precalculus course is built to deliver a comprehensive, illuminating, engaging, and Common Core aligned experience!