Functions

# Transformation of Functions

### Learning Objectives

In this section, you will:

• Graph functions using vertical and horizontal shifts.
• Graph functions using reflections about the and the
• 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 functionthe functionis shifted verticallyunits. See (Figure) for an example.

To help you visualize the concept of a vertical shift, consider thatTherefore,is equivalent toEvery unit ofis replaced byso the y-value increases or decreases depending on the value ofThe result is a shift upward or downward.

### Vertical Shift

Given a function a new function where is a constant, is a vertical shift of the function All the output values change by units. If is positive, the graph will shift up. If 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) shows the area of open vents(in square feet) throughout the day in hours after midnight,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).

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

This notation tells us that, for any value ofcan be found by evaluating the functionat the same input and then adding 20 to the result. This definesas a transformation of the functionin 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 (Figure).

 0 8 10 17 19 24 0 0 220 220 0 0 20 20 240 240 20 20

### How To

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

1. Identify the output row or column.
2. Determine the magnitude of the shift.
3. 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 functionis given in (Figure). Create a table for the function

 2 4 6 8 1 3 7 11
[hidden-answer a=”281112″]The formulatells us that we can find the output values ofby subtracting 3 from the output values ofFor example:

#### Analysis

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

### Try It

The functiongives the heightof a ball (in meters) thrown upward from the ground afterseconds. Suppose the ball was instead thrown from the top of a 10-m building. Relate this new height functiontoand then find a formula for

#### 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).

For example, ifthenis 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

### Horizontal Shift

Given a functiona new functionwhereis a constant, is a horizontal shift of the functionIfis positive, the graph will shift right. Ifis negative, the graph will shift left.

### Adding a Constant to an Input

Returning to our building airflow example from (Figure), 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 setto be the original program andto be the revised program.

In the new graph, at each time, the airflow is the same as the original functionwas 2 hours later. For example, in the original functionthe airflow starts to change at 8 a.m., whereas for the functionthe airflow starts to change at 6 a.m. The comparable function values areSee (Figure). Notice also that the vents first opened toat 10 a.m. under the original plan, while under the new plan the vents reachat

In both cases, we see that, becausestarts 2 hours sooner,That means that the same output values are reached when

#### Analysis

Note thathas 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 functionuses the same outputs asbut matches those outputs to inputs 2 hours earlier than those ofSaid another way, we must add 2 hours to the input ofto find the corresponding output for

### How To

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

1. Identify the input row or column.
2. Determine the magnitude of the shift.
3. Add the shift to the value in each input cell.

### Shifting a Tabular Function Horizontally

A functionis given in (Figure). Create a table for the function

 2 4 6 8 1 3 7 11

The formulatells us that the output values ofare the same as the output value of when the input value is 3 less than the original value. For example, we know thatTo get the same output from the functionwe will need an input value that is 3 larger. We input a value that is 3 larger forbecause the function takes 3 away before evaluating the function

We continue with the other values to create (Figure).

 5 7 9 11 2 4 6 8 1 3 7 11 1 3 7 11

The result is that the functionhas been shifted to the right by 3. Notice the output values forremain the same as the output values forbut the corresponding input values,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.

#### Analysis

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

### Identifying a Horizontal Shift of a Toolkit Function

(Figure) represents a transformation of the toolkit functionRelate this new functiontoand then find a formula for

Notice that the graph is identical in shape to thefunction, 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 valueto get the output valuethe x-values must be 2 units larger because of the shift to the right by 2 units. We can then use the definition of thefunction to write a formula forby evaluating

#### Analysis

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

### Interpreting Horizontal versus Vertical Shifts

The functiongives the number of gallons of gas required to drivemiles. Interpretand

can be interpreted as adding 10 to the output, gallons. This is the gas required to drivemiles, plus another 10 gallons of gas. The graph would indicate a vertical shift.

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 thanmiles. The graph would indicate a horizontal shift.

### Try It

Given the functiongraph the original functionand the transformationon the same axes. Is this a horizontal or a vertical shift? Which way is the graph shifted and by how many units?

The graphs ofandare shown below. The transformation is a horizontal shift. The function is shifted to the left by 2 units.

#### Combining Vertical and Horizontal Shifts

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

### How To

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

1. Identify the vertical and horizontal shifts from the formula.
2. 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.
3. 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.
4. Apply the shifts to the graph in either order.

### Graphing Combined Vertical and Horizontal Shifts

Givensketch a graph of

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

Let us follow one point of the graph of

• The pointis transformed first by shifting left 1 unit:
• The pointis transformed next by shifting down 3 units:

(Figure) shows the graph of

### Try It

Givensketch a graph of

### Identifying Combined Vertical and Horizontal Shifts

Write a formula for the graph shown in (Figure), 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

#### Analysis

Note that this transformation has changed the domain and range of the function. This new graph has domainand range

### Try It

Write a formula for a transformation of the toolkit reciprocal functionthat 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).

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 a new functionis a vertical reflection of the functionsometimes called a reflection about (or over, or through) the x-axis.

Given a functiona new functionis a horizontal reflection of the functionsometimes called a reflection about the y-axis.

### How To

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

1. Multiply all outputs by –1 for a vertical reflection. The new graph is a reflection of the original graph about the x-axis.
2. 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 (a) vertically and (b) horizontally.

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

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 outputvalues, so the negative sign belongs outside of the function.

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

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 domainand rangethe vertical reflection gives thefunction the range and the horizontal reflection gives thefunction the domain

### Try It

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

### Reflecting a Tabular Function Horizontally and Vertically

A functionis given as (Figure). Create a table for the functions below.

 2 4 6 8 1 3 7 11

1. Forthe 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 (Figure).

 2 4 6 8 –1 –3 –7 –11
2. Forthe negative sign inside the function indicates a horizontal reflection, so each input value will be the opposite of the original input value and thevalues stay the same as thevalues. See (Figure).

 −2 −4 −6 −8 1 3 7 11

### Try It

A functionis given as (Figure). Create a table for the functions below.

 −2 0 2 4 5 10 15 20

1.  -2 0 2 4
2.  -2 0 2 4 15 10 5 unknown

### Applying a Learning Model Equation

A common model for learning has an equation similar to where is the percentage of mastery that can be achieved after practice sessions. This is a transformation of the function shown in (Figure). Sketch a graph of

This equation combines three transformations into one equation.

• A horizontal reflection:
• A vertical reflection:
• A vertical shift:

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).

1. First, we apply a horizontal reflection: (0, 1) (–1, 2).
2. Then, we apply a vertical reflection: (0, -1) (-1, –2)
3. 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), 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.

#### Analysis

As a model for learning, this function would be limited to a domain ofwith corresponding range

### Try It

Given the toolkit functiongraphandTake note of any surprising behavior for these functions.

Notice:looks the same as.

### Determining Even and Odd Functions

Some functions exhibit symmetry so that reflections result in the original graph. For example, horizontally reflecting the toolkit functions or will result in the original graph. We say that these types of graphs are symmetric about the y-axis. A function whose graph is symmetric about the y-axis is called an even function.

If the graphs oforwere reflected over both axes, the result would be the original graph, as shown in (Figure).

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,is neither even nor odd. Also, the only function that is both even and odd is the constant function

### Even and Odd Functions

A function is called an even function if for every input

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

A function is called an odd function if for every input

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

### How To

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

1. Determine whether the function satisfiesIf it does, it is even.
2. Determine whether the function satisfiesIf it does, it is odd.
3. 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 functioneven, 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.

Becausethis is an odd function.

#### Analysis

Consider the graph ofin (Figure). Notice that the graph is symmetric about the origin. For every pointon the graph, the corresponding pointis also on the graph. For example, (1, 3) is on the graph ofand the corresponding pointis also on the graph.

### Try It

Is the functioneven, odd, or neither?

even

### 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) shows a function multiplied by constant factors 2 and 0.5 and the resulting vertical stretch and compression.

### Vertical Stretches and Compressions

Given a functiona new functionwhereis a constant, is a vertical stretch or vertical compression of the function

• Ifthen the graph will be stretched.
• Ifthen the graph will be compressed.
• Ifthen there will be combination of a vertical stretch or compression with a vertical reflection.

### How To

Given a function, graph its vertical stretch.

1. Identify the value of
2. Multiply all range values by
3. Ifthe graph is stretched by a factor of

Ifthe graph is compressed by a factor of

Ifthe graph is either stretched or compressed and also reflected about the x-axis.

### Graphing a Vertical Stretch

A functionmodels the population of fruit flies. The graph is shown in (Figure).

A scientist is comparing this population to another population,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).

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 inputthe value of the functionis twice the value of the functionNotice 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,stay the same while the output values are twice as large as before.[/hidden-answer]

### How To

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

1. Determine the value of
2. Multiply all of the output values by

### Finding a Vertical Compression of a Tabular Function

A functionis given as (Figure). Create a table for the function

 2 4 6 8 1 3 7 11

The formulatells us that the output values ofare half of the output values ofwith the same inputs. For example, we know thatThen

We do the same for the other values to produce (Figure).

#### Analysis

The result is that the functionhas been compressed vertically byEach output value is divided in half, so the graph is half the original height.

### Try It

A functionis given as (Figure). Create a table for the function

 2 4 6 8 12 16 20 0

 2 4 6 8 9 12 15 0

### Recognizing a Vertical Stretch

The graph in (Figure) is a transformation of the toolkit functionRelate this new functiontoand then find a formula for

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 thatWith the basic cubic function at the same input, Based on that, it appears that the outputs ofarethe outputs of the functionbecauseFrom this we can fairly safely conclude that

We can write a formula forby using the definition of the function

### Try It

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 functionthe formresults in a horizontal stretch or compression. Consider the functionObserve (Figure). The graph ofis a horizontal stretch of the graph of the functionby a factor of 2. The graph ofis a horizontal compression of the graph of the functionby a factor of 2.

### Horizontal Stretches and Compressions

Given a functiona new functionwhereis a constant, is a horizontal stretch or horizontal compression of the function

• Ifthen the graph will be compressed by
• Ifthen the graph will be stretched by
• Ifthen there will be combination of a horizontal stretch or compression with a horizontal reflection.

### How To

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

1. Write a formula to represent the function.
2. Setwherefor a compression or
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,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) for a graphical comparison of the original population and the compressed population.

#### Analysis

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 functionis given as (Figure). Create a table for the function

 2 4 6 8 1 3 7 11

The formulatells us that the output values forare the same as the output values for the functionat an input half the size. Notice that we do not have enough information to determinebecauseand we do not have a value forin our table. Our input values towill need to be twice as large to get inputs forthat we can evaluate. For example, we can determine

We do the same for the other values to produce (Figure).

 4 8 12 16 1 3 7 11

(Figure) shows the graphs of both of these sets of points.

#### Analysis

Because each input value has been doubled, the result is that the functionhas been stretched horizontally by a factor of 2.

### Recognizing a Horizontal Compression on a Graph

Relate the functiontoin (Figure).

[hidden-answer a=”fs-id1165137737597″]The graph oflooks like the graph ofhorizontally compressed. Becauseends atandends atwe can see that the values have been compressed bybecauseWe might also notice thatandEither way, we can describe this relationship asThis is a horizontal compression by[/hidden-answer]

#### Analysis

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 ofin our function: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.

### Try It

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

so using the square root function we get

### 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 aswhich transformation should we start with? The answer here follows nicely from the order of operations. Given the output value ofwe 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 writefor example, we have to think about how the inputs to the functionrelate to the inputs to the functionSuppose we knowWhat input towould produce that output? In other words, what value ofwill allowWe would needTo solve forwe 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 formfirst vertically stretch byand then vertically shift by

When combining horizontal transformations written in the formfirst horizontally shift byand then horizontally stretch by

When combining horizontal transformations written in the formfirst horizontally stretch byand then horizontally shift by

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 (Figure) for the functioncreate a table of values for the function

 6 12 18 24 10 14 15 17

There are three steps to this transformation, and we will work from the inside out. Starting with the horizontal transformations,is a horizontal compression bywhich means we multiply each value bySee (Figure).

 2 4 6 8 10 14 15 17

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 (Figure).

 2 4 6 8 20 28 30 34

Finally, we can apply the vertical shift, which will add 1 to all the output values. See (Figure).

 2 4 6 8 21 29 31 35

### Finding a Triple Transformation of a Graph

Use the graph ofin (Figure) to sketch a graph of

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

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).

Next, we horizontally shift left by 2 units, as indicated bySee (Figure).

Last, we vertically shift down by 3 to complete our sketch, as indicated by theon the outside of the function. See (Figure).][/hidden-answer]

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

### Key Equations

 Vertical shift (up for) Horizontal shift (right for) Vertical reflection Horizontal reflection Vertical stretch ( ) Vertical compression [latex]\left(0

### Key Concepts

• A function can be shifted vertically by adding a constant to the output. See (Figure) and (Figure).
• A function can be shifted horizontally by adding a constant to the input. See (Figure), (Figure), and (Figure).
• Relating the shift to the context of a problem makes it possible to compare and interpret vertical and horizontal shifts. See (Figure).
• Vertical and horizontal shifts are often combined. See (Figure) and (Figure).
• A vertical reflection reflects a graph about the axis. A graph can be reflected vertically by multiplying the output by –1.
• A horizontal reflection reflects a graph about the axis. A graph can be reflected horizontally by multiplying the input by –1.
• A graph can be reflected both vertically and horizontally. The order in which the reflections are applied does not affect the final graph. See (Figure).
• A function presented in tabular form can also be reflected by multiplying the values in the input and output rows or columns accordingly. See (Figure).
• A function presented as an equation can be reflected by applying transformations one at a time. See (Figure).
• Even functions are symmetric about the axis, whereas odd functions are symmetric about the origin.
• Even functions satisfy the condition
• Odd functions satisfy the condition
• A function can be odd, even, or neither. See (Figure).
• A function can be compressed or stretched vertically by multiplying the output by a constant. See (Figure), (Figure), and (Figure).
• A function can be compressed or stretched horizontally by multiplying the input by a constant. See (Figure), (Figure), and (Figure).
• The order in which different transformations are applied does affect the final function. Both vertical and horizontal transformations must be applied in the order given. However, a vertical transformation may be combined with a horizontal transformation in any order. See (Figure) and (Figure).

### Section Exercises

#### Verbal

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?

A horizontal shift results when a constant is added to or subtracted from the input. A vertical shifts results when a constant is added to or subtracted from the output.

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?

A horizontal compression results when a constant greater than 1 is multiplied by the input. A vertical compression results when a constant between 0 and 1 is multiplied by the output.

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?

For a functionsubstituteforinSimplify. If the resulting function is the same as the original function,then the function is even. If the resulting function is the opposite of the original function,then the original function is odd. If the function is not the same or the opposite, then the function is neither odd nor even.

#### Algebraic

For the following exercises, write a formula for the function obtained when the graph is shifted as described.

is shifted up 1 unit and to the left 2 units.

is shifted down 3 units and to the right 1 unit.

is shifted down 4 units and to the right 3 units.

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

The graph ofis a horizontal shift to the left 43 units of the graph of

The graph ofis a horizontal shift to the right 4 units of the graph of

The graph ofis a vertical shift up 8 units of the graph of

The graph ofis a vertical shift down 7 units of the graph of

The graph of is a horizontal shift to the left 4 units and a vertical shift down 1 unit of the graph of

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

decreasing onand increasing on

decreasing on

#### Graphical

For the following exercises, use the graph ofshown in (Figure) to sketch a graph of each transformation of

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

#### Numeric

Tabular representations for the functionsandare given below. Writeandas transformations of

 −2 −1 0 1 2 −2 −1 −3 1 2
 −1 0 1 2 3 −2 −1 −3 1 2
 −2 −1 0 1 2 −1 0 −2 2 3

Tabular representations for the functionsandare given below. Writeandas transformations of

 −2 −1 0 1 2 −1 −3 4 2 1
 −3 −2 −1 0 1 −1 −3 4 2 1
 −2 −1 0 1 2 −2 −4 3 1 0

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.

even

odd

even

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

The graph ofis a vertical reflection (across the -axis) of the graph of

The graph ofis a vertical stretch by a factor of 4 of the graph of

The graph ofis a horizontal compression by a factor ofof the graph of

The graph ofis a horizontal stretch by a factor of 3 of the graph of

The graph ofis a horizontal reflection across the-axis and a vertical stretch by a factor of 3 of the graph of

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

The graph ofis reflected over theaxis and horizontally compressed by a factor of
.

The graph ofis reflected over the-axis and horizontally stretched by a factor of 2.

The graph ofis vertically compressed by a factor ofthen shifted to the left 2 units and down 3 units.

The graph ofis vertically stretched by a factor of 8, then shifted to the right 4 units and up 2 units.

The graph ofis vertically compressed by a factor ofthen shifted to the right 5 units and up 1 unit.

The graph ofis 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.

The graph of the functionis shifted to the left 1 unit, stretched vertically by a factor of 4, and shifted down 5 units.

The graph ofis stretched vertically by a factor of 2, shifted horizontally 4 units to the right, reflected across the horizontal axis, and then shifted vertically 3 units up.

The graph of the functionis compressed vertically by a factor of

The graph of the function is stretched horizontally by a factor of 3 and then shifted vertically downward by 3 units.

The graph ofis shifted right 4 units and then reflected across the vertical line

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

### Glossary

even function
a function whose graph is unchanged by horizontal reflection,and is symmetric about the axis
horizontal compression
a transformation that compresses a function’s graph horizontally, by multiplying the input by a constant
horizontal reflection
a transformation that reflects a function’s graph across the y-axis by multiplying the input by
horizontal shift
a transformation that shifts a function’s graph left or right by adding a positive or negative constant to the input
horizontal stretch
a transformation that stretches a function’s graph horizontally by multiplying the input by a constant
odd function
a function whose graph is unchanged by combined horizontal and vertical reflection,and is symmetric about the origin
vertical compression
a function transformation that compresses the function’s graph vertically by multiplying the output by a constant
vertical reflection
a transformation that reflects a function’s graph across the x-axis by multiplying the output by
vertical shift
a transformation that shifts a function’s graph up or down by adding a positive or negative constant to the output
vertical stretch
a transformation that stretches a function’s graph vertically by multiplying the output by a constant