Again Assuming a Transformation From F(T)=-6t^t

LEARNING OBJECTIVES

By the finish of this lesson, you volition be able to:

• Graph functions using vertical and horizontal shifts.
• Graph functions using reflections about the [latex]x[/latex] -axis and the [latex]y[/latex] -axis.
• Decide whether a office is even, odd, or neither from its graph.
• Graph functions using compressions and stretches.
• Combine transformations.

Effigy 1.(credit: "Misko"/Flickr)

We all know that a apartment 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. Just what happens when we bend a flexible mirror? Similar a carnival funhouse mirror, it presents us with a distorted epitome of ourselves, stretched or compressed horizontally or vertically. In a similar way, we tin distort or transform mathematical functions to meliorate adapt them to describing objects or processes in the real world. In this section, nosotros will take a wait 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 course 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 modify functions to construct appropriate models for the bug nosotros are trying to solve.

Identifying Vertical Shifts

One simple kind of transformation involves shifting the unabridged graph of a function up, down, correct, or left. The simplest shift is a vertical shift, moving the graph upwardly or down, because this transformation involves adding a positive or negative constant to the role. In other words, we add together the same constant to the output value of the function regardless of the input. For a function [latex]g\left(ten\correct)=f\left(ten\correct)+k[/latex], the function [latex]f\left(x\right)[/latex] is shifted vertically [latex]thousand[/latex] units.

Effigy two. Vertical shift by [latex]m=1[/latex] of the cube root function [latex]f\left(x\correct)=\sqrt[3]{x}[/latex].

To help you lot visualize the concept of a vertical shift, consider that [latex]y=f\left(x\correct)[/latex]. Therefore, [latex]f\left(ten\right)+one thousand[/latex] is equivalent to [latex]y+yard[/latex]. Every unit of measurement of [latex]y[/latex] is replaced past [latex]y+k[/latex], so the [latex]y\text{-}[/latex] value increases or decreases depending on the value of [latex]one thousand[/latex]. The consequence is a shift upward or downward.

A General Note: Vertical Shift

Given a function [latex]f\left(ten\correct)[/latex], a new function [latex]m\left(ten\right)=f\left(x\right)+yard[/latex], where [latex]chiliad[/latex] is a abiding, is a vertical shift of the part [latex]f\left(ten\right)[/latex]. All the output values modify by [latex]thou[/latex] units. If [latex]k[/latex] is positive, the graph will shift up. If [latex]k[/latex] is negative, the graph will shift downwardly.

Example 1: 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 2 shows the area of open vents [latex]V[/latex] (in foursquare anxiety) throughout the twenty-four hour period in hours later midnight, [latex]t[/latex]. During the summertime, the facilities manager decides to try to meliorate regulate temperature by increasing the corporeality of open up vents by xx square feet throughout the 24-hour interval and nighttime. Sketch a graph of this new part.

Effigy 3

Solution

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

Figure iv

Discover that for each input value, the output value has increased by 20, so if we call the new role [latex]Southward\left(t\right)[/latex], we could write

[latex]Southward\left(t\right)=5\left(t\right)+20[/latex]

This annotation tells us that, for whatsoever value of [latex]t,S\left(t\right)[/latex] can be establish by evaluating the part [latex]V[/latex] at the same input so adding 20 to the result. This defines [latex]S[/latex] as a transformation of the function [latex]Five[/latex], in this case a vertical shift up xx units. Notice that, with a vertical shift, the input values stay the same and only the output values change.

[latex]t[/latex]	0	8	10	17	nineteen	24
[latex]V\left(t\right)[/latex]	0	0	220	220	0	0
[latex]S\left(t\right)[/latex]	xx	20	240	240	20	twenty

How To: Given a tabular function, create a new row to correspond 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 jail cell. Add a positive value for up or a negative value for down.

Example 2: Shifting a Tabular Part Vertically

A role [latex]f\left(ten\right)[/latex] is given below. Create a table for the office [latex]thousand\left(x\right)=f\left(x\right)-3[/latex].

[latex]x[/latex]	2	4	6	8
[latex]f\left(10\right)[/latex]	1	3	vii	eleven

Solution

The formula [latex]thou\left(ten\right)=f\left(x\right)-3[/latex] tells us that nosotros tin detect the output values of [latex]chiliad[/latex] by subtracting 3 from the output values of [latex]f[/latex]. For case:

[latex]\begin{cases}f\left(two\right)=one\hfill & \text{Given}\hfill \\ g\left(10\right)=f\left(x\correct)-3\hfill & \text{Given transformation}\hfill \\ g\left(2\right)=f\left(2\right)-3\hfill & \hfill \\ =i - 3\hfill & \hfill \\ =-2\hfill & \hfill \finish{cases}[/latex]

Subtracting 3 from each [latex]f\left(x\right)[/latex] value, we can complete a table of values for [latex]g\left(ten\right)[/latex].

[latex]ten[/latex]	2	4	6	viii
[latex]f\left(x\right)[/latex]	1	three	vii	eleven
[latex]thou\left(10\correct)[/latex]	−2	0	4	viii

The part [latex]h\left(t\right)=-4.nine{t}^{2}+30t[/latex] gives the height [latex]h[/latex] of a brawl (in meters) thrown upward from the basis later [latex]t[/latex] seconds. Suppose the ball was instead thrown from the top of a 10-thou building. Relate this new summit function [latex]b\left(t\right)[/latex] to [latex]h\left(t\correct)[/latex], and so find a formula for [latex]b\left(t\right)[/latex].

[latex]b\left(t\right)=h\left(t\right)+10=-4.9{t}^{two}+30t+x[/latex]

Identifying Horizontal Shifts

Nosotros 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, modify its graph and significant. A shift to the input results in a movement of the graph of the office left or right in what is known as a horizontal shift.

Figure v. Horizontal shift of the function [latex]f\left(10\right)=\sqrt[iii]{10}[/latex]. Note that [latex]h=+1[/latex] shifts the graph to the left, that is, towards negative values of [latex]x[/latex].

For example, if [latex]f\left(ten\right)={x}^{ii}[/latex], so [latex]grand\left(x\right)={\left(ten - ii\right)}^{ii}[/latex] is a new function. Each input is reduced by two prior to squaring the function. The result is that the graph is shifted two units to the correct, because nosotros would need to increase the prior input past 2 units to yield the same output value as given in [latex]f[/latex].

A General Note: Horizontal Shift

Given a function [latex]f[/latex], a new role [latex]one thousand\left(x\right)=f\left(10-h\correct)[/latex], where [latex]h[/latex] is a constant, is a horizontal shift of the part [latex]f[/latex]. If [latex]h[/latex] is positive, the graph will shift correct. If [latex]h[/latex] is negative, the graph volition shift left.

Example iii: Calculation a Constant to an Input

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

Solution

We can gear up [latex]V\left(t\right)[/latex] to be the original program and [latex]F\left(t\right)[/latex] to be the revised plan.

[latex]\brainstorm{cases}{c}V\left(t\right)=\text{ the original venting programme}\\ \text{F}\left(t\right)=\text{starting 2 hrs sooner}\stop{cases}[/latex]

In the new graph, at each time, the airflow is the aforementioned as the original function [latex]Five[/latex] was 2 hours later. For example, in the original part [latex]V[/latex], the airflow starts to change at 8 a.m., whereas for the office [latex]F[/latex], the airflow starts to change at half-dozen a.1000. The comparable part values are [latex]V\left(viii\right)=F\left(half-dozen\right)[/latex]. Notice also that the vents first opened to [latex]220{\text{ ft}}^{2}[/latex] at 10 a.m. under the original plan, while under the new programme the vents achieve [latex]220{\text{ ft}}^{\text{2}}[/latex] at eight a.one thousand., so [latex]5\left(10\right)=F\left(viii\correct)[/latex].

Figure 6

In both cases, we see that, because [latex]F\left(t\right)[/latex] starts 2 hours sooner, [latex]h=-2[/latex]. That means that the aforementioned output values are reached when [latex]F\left(t\right)=V\left(t-\left(-2\right)\right)=V\left(t+two\right)[/latex].

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.

Example 4: Shifting a Tabular Office Horizontally

A function [latex]f\left(x\right)[/latex] is given below. Create a table for the role [latex]g\left(10\right)=f\left(x - three\right)[/latex].

[latex]x[/latex]	2	4	vi	8
[latex]f\left(x\correct)[/latex]	i	3	seven	11

Solution

The formula [latex]k\left(x\right)=f\left(x - 3\right)[/latex] tells usa that the output values of [latex]thou[/latex] are the same equally the output value of [latex]f[/latex] when the input value is 3 less than the original value. For example, we know that [latex]f\left(2\correct)=one[/latex]. To go the same output from the function [latex]yard[/latex], we will demand an input value that is iii larger. We input a value that is 3 larger for [latex]m\left(ten\right)[/latex] considering the role takes iii away before evaluating the office [latex]f[/latex].

[latex]\brainstorm{cases}grand\left(5\right)=f\left(five - three\right)\hfill \\ =f\left(2\correct)\hfill \\ =1\hfill \end{cases}[/latex]

We continue with the other values to create this table.

[latex]x[/latex]	five	seven	9	11
[latex]x - three[/latex]	2	4	6	8
[latex]f\left(10\right)[/latex]	i	three	7	11
[latex]g\left(ten\right)[/latex]	ane	three	seven	eleven

The result is that the function [latex]chiliad\left(x\right)[/latex] has been shifted to the correct by 3. Notice the output values for [latex]g\left(x\correct)[/latex] remain the same as the output values for [latex]f\left(x\right)[/latex], but the corresponding input values, [latex]x[/latex], have shifted to the right by 3. Specifically, ii shifted to five, 4 shifted to seven, vi shifted to nine, and eight shifted to 11.

Case 5: Identifying a Horizontal Shift of a Toolkit Role

This graph represents a transformation of the toolkit office [latex]f\left(x\correct)={x}^{2}[/latex]. Relate this new function [latex]g\left(x\right)[/latex] to [latex]f\left(x\right)[/latex], and then find a formula for [latex]g\left(x\right)[/latex].

Figure eight

Solution

Notice that the graph is identical in shape to the [latex]f\left(x\right)={x}^{two}[/latex] part, but the x-values are shifted to the right 2 units. The vertex used to be at (0,0), just now the vertex is at (2,0). The graph is the basic quadratic part shifted two units to the correct, and so

[latex]m\left(x\correct)=f\left(x - 2\right)[/latex]

Detect how nosotros must input the value [latex]x=ii[/latex] to get the output value [latex]y=0[/latex]; the x-values must exist ii units larger because of the shift to the right past two units. Nosotros can then use the definition of the [latex]f\left(10\right)[/latex] function to write a formula for [latex]g\left(x\correct)[/latex] by evaluating [latex]f\left(x - 2\right)[/latex].

[latex]\begin{cases}f\left(10\right)={x}^{2}\hfill \\ m\left(ten\right)=f\left(x - 2\right)\hfill \\ yard\left(ten\right)=f\left(x - 2\right)={\left(x - 2\right)}^{2}\hfill \cease{cases}[/latex]

Example 6: Interpreting Horizontal versus Vertical Shifts

The role [latex]G\left(chiliad\right)[/latex] gives the number of gallons of gas required to bulldoze [latex]m[/latex] miles. Translate [latex]G\left(k\correct)+10[/latex] and [latex]Chiliad\left(1000+10\right)[/latex].

Solution

[latex]G\left(m\right)+10[/latex] tin be interpreted every bit adding x to the output, gallons. This is the gas required to bulldoze [latex]1000[/latex] miles, plus some other 10 gallons of gas. The graph would point a vertical shift.

[latex]Chiliad\left(m+10\right)[/latex] can be interpreted equally adding 10 to the input, miles. So this is the number of gallons of gas required to bulldoze ten miles more than than [latex]1000[/latex] miles. The graph would indicate a horizontal shift.

Try It one

Given the part [latex]f\left(x\right)=\sqrt{10}[/latex], graph the original role [latex]f\left(x\right)[/latex] and the transformation [latex]g\left(x\right)=f\left(10+ii\right)[/latex] on the same axes. Is this a horizontal or a vertical shift? Which way is the graph shifted and by how many units?

Solution

Graphing Functions Using Reflections about the Axes

Another transformation that tin can be applied to a function is a reflection over the 10– or y-axis. A vertical reflection reflects a graph vertically beyond the x-axis, while a horizontal reflection reflects a graph horizontally beyond the y-centrality. The reflections are shown in Figure 9.

Effigy 9. Vertical and horizontal reflections of a function.

Notice that the vertical reflection produces a new graph that is a mirror image of the base or original graph nearly the 10-axis. The horizontal reflection produces a new graph that is a mirror image of the base or original graph about the y-axis.

A General Note: Reflections

Given a function [latex]f\left(ten\right)[/latex], a new function [latex]yard\left(10\right)=-f\left(x\right)[/latex] is a vertical reflection of the function [latex]f\left(x\right)[/latex], sometimes chosen a reflection about (or over, or through) the x-centrality.

Given a function [latex]f\left(x\right)[/latex], a new function [latex]g\left(ten\right)=f\left(-x\right)[/latex] is a horizontal reflection of the function [latex]f\left(x\right)[/latex], sometimes called a reflection about the y-axis.

How To: Given a role, reflect the graph both vertically and horizontally.

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

Example 7: Reflecting a Graph Horizontally and Vertically

Reflect the graph of [latex]due south\left(t\right)=\sqrt{t}[/latex] (a) vertically and (b) horizontally.

Solution

a. Reflecting the graph vertically means that each output value will exist reflected over the horizontal t-axis as shown in Figure 10.

Effigy x.Vertical reflection of the foursquare root part

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

[latex]V\left(t\right)=-s\left(t\right)\text{ or }V\left(t\right)=-\sqrt{t}[/latex]

Notice that this is an exterior modify, or vertical shift, that affects the output [latex]s\left(t\correct)[/latex] values, so the negative sign belongs exterior of the function.

b.

Reflecting horizontally means that each input value will exist reflected over the vertical centrality as shown in Figure 11.

Effigy 11. Horizontal reflection of the foursquare root function

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

[latex]H\left(t\right)=s\left(-t\correct)\text{ or }H\left(t\right)=\sqrt{-t}[/latex]

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

Note that these transformations tin affect the domain and range of the functions. While the original foursquare root function has domain [latex]\left[0,\infty \correct)[/latex] and range [latex]\left[0,\infty \right)[/latex], the vertical reflection gives the [latex]Five\left(t\correct)[/latex] part the range [latex]\left(-\infty ,0\right][/latex] and the horizontal reflection gives the [latex]H\left(t\right)[/latex] function the domain [latex]\left(-\infty ,0\correct][/latex].

Try It 2

Reflect the graph of [latex]f\left(x\correct)=|ten - i|[/latex] (a) vertically and (b) horizontally.

Solution

Example 8: Reflecting a Tabular Office Horizontally and Vertically

A function [latex]f\left(x\right)[/latex] is given. Create a table for the functions below.

1. [latex]g\left(x\right)=-f\left(x\right)[/latex]
2. [latex]h\left(ten\right)=f\left(-x\correct)[/latex]

[latex]x[/latex]	ii	4	six	viii
[latex]f\left(x\right)[/latex]	1	three	7	eleven

Solution

1. For [latex]chiliad\left(10\right)[/latex], 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.

   [latex]x[/latex]	2	4	six	eight
   [latex]m\left(x\right)[/latex]	–one	–three	–seven	–eleven

2. For [latex]h\left(10\right)[/latex], the negative sign inside the role indicates a horizontal reflection, so each input value volition exist the opposite of the original input value and the [latex]h\left(10\correct)[/latex] values stay the same as the [latex]f\left(x\right)[/latex] values.

   [latex]x[/latex]	−2	−four	−6	−8
   [latex]h\left(10\right)[/latex]	1	iii	7	11

Attempt It iii

[latex]10[/latex]	−2	0	2	4
[latex]f\left(x\right)[/latex]	5	10	15	twenty

Using the function [latex]f\left(10\right)[/latex] given in the table above, create a table for the functions below.

a. [latex]g\left(x\correct)=-f\left(x\right)[/latex]

b. [latex]h\left(10\right)=f\left(-x\right)[/latex]

Solution

Determining Even and Odd Functions

Some functions exhibit symmetry then that reflections result in the original graph. For example, horizontally reflecting the toolkit functions [latex]f\left(x\correct)={10}^{2}[/latex] or [latex]f\left(x\right)=|10|[/latex] will result in the original graph. We say that these types of graphs are symmetric about the y-centrality. Functions whose graphs are symmetric about the y-axis are called even functions.

If the graphs of [latex]f\left(x\correct)={x}^{iii}[/latex] or [latex]f\left(x\correct)=\frac{i}{x}[/latex] were reflected over both axes, the consequence would be the original graph.

Figure 12. (a) The cubic toolkit function (b) Horizontal reflection of the cubic toolkit part (c) Horizontal and vertical reflections reproduce the original cubic role.

We say that these graphs are symmetric almost the origin. A office with a graph that is symmetric near the origin is called an odd role.

Note: A function can be neither even nor odd if it does not exhibit either symmetry. For case, [latex]f\left(x\correct)={2}^{10}[/latex] is neither even nor odd. Also, the only function that is both even and odd is the constant function [latex]f\left(x\correct)=0[/latex].

A General Note: Even and Odd Functions

A role is called an fifty-fifty function if for every input [latex]x[/latex]

[latex]f\left(x\right)=f\left(-x\correct)[/latex]

The graph of an even function is symmetric about the [latex]y\text{-}[/latex] centrality.

A function is chosen an odd function if for every input [latex]x[/latex]

[latex]f\left(x\right)=-f\left(-x\right)[/latex]

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

How To: Given the formula for a function, determine if the office is even, odd, or neither.

1. Determine whether the role satisfies [latex]f\left(x\right)=f\left(-x\correct)[/latex]. If information technology does, it is even.
2. Decide whether the role satisfies [latex]f\left(x\right)=-f\left(-x\right)[/latex]. If information technology does, it is odd.
3. If the function does not satisfy either rule, it is neither even nor odd.

Example 9: Determining whether a Function Is Even, Odd, or Neither

Is the function [latex]f\left(x\right)={x}^{three}+2x[/latex] even, odd, or neither?

Solution

Without looking at a graph, nosotros tin can determine whether the office is fifty-fifty or odd past finding formulas for the reflections and determining if they render us to the original role. Let's begin with the rule for even functions.

[latex]f\left(-x\right)={\left(-x\right)}^{3}+2\left(-x\right)=-{x}^{3}-2x[/latex]

This does not return united states to the original function, then this function is not even. Nosotros tin can now test the dominion for odd functions.

[latex]-f\left(-x\right)=-\left(-{x}^{3}-2x\right)={x}^{3}+2x[/latex]

Because [latex]-f\left(-x\correct)=f\left(x\right)[/latex], this is an odd function.

Try It 4

Is the role [latex]f\left(s\correct)={s}^{4}+three{s}^{2}+seven[/latex] even, odd, or neither?

Solution

Graphing Functions Using Stretches and Compressions

Calculation a abiding to the inputs or outputs of a function changed the position of a graph with respect to the axes, just it did not bear on the shape of a graph. Nosotros now explore the effects of multiplying the inputs or outputs by some quantity.

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

Vertical Stretches and Compressions

When we multiply a part by a positive abiding, 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 ane, nosotros get a vertical stretch; if the constant is between 0 and one, we go a vertical compression. The graph below shows a function multiplied by constant factors two and 0.5 and the resulting vertical stretch and compression.

Figure fourteen. Vertical stretch and compression

A General Note: Vertical Stretches and Compressions

Given a role [latex]f\left(x\right)[/latex], a new part [latex]g\left(ten\right)=af\left(x\right)[/latex], where [latex]a[/latex] is a constant, is a vertical stretch or vertical pinch of the part [latex]f\left(x\correct)[/latex].

• If [latex]a>1[/latex], then the graph will be stretched.
• If 0 < a < 1, then the graph volition exist compressed.
• If [latex]a<0[/latex], then at that place 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 [latex]a[/latex].
2. Multiply all range values by [latex]a[/latex].

3. If [latex]a>1[/latex], the graph is stretched by a cistron of [latex]a[/latex].

   If [latex]{ 0 }<{ a }<{ i }[/latex], the graph is compressed by a factor of [latex]a[/latex].

   If [latex]a<0[/latex], the graph is either stretched or compressed and besides reflected nearly the x-axis.

Case x: Graphing a Vertical Stretch

Figure 15

A function [latex]P\left(t\right)[/latex] models the population of fruit flies.

A scientist is comparing this population to some other population, [latex]Q[/latex], whose growth follows the aforementioned pattern, simply is twice equally large. Sketch a graph of this population.

Solution

Considering the population is e'er twice as large, the new population'southward output values are always twice the original part'due south output values.

If we choose four reference points, (0, 1), (3, three), (6, ii) and (vii, 0) we will multiply all of the outputs by two.

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

[latex]\brainstorm{cases}\left(0,\text{ }1\right)\to \left(0,\text{ }2\right)\hfill \\ \left(three,\text{ }iii\right)\to \left(iii,\text{ }vi\right)\hfill \\ \left(vi,\text{ }2\right)\to \left(6,\text{ }4\right)\hfill \\ \left(7,\text{ }0\correct)\to \left(7,\text{ }0\right)\hfill \stop{cases}[/latex]

Effigy sixteen

Symbolically, the relationship is written every bit

[latex]Q\left(t\right)=2P\left(t\right)[/latex]

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

How To: Given a tabular office and assuming that the transformation is a vertical stretch or pinch, create a tabular array for a vertical pinch.

1. Decide the value of [latex]a[/latex].
2. Multiply all of the output values by [latex]a[/latex].

Example 11: Finding a Vertical Compression of a Tabular Role

A office [latex]f[/latex] is given in the table below. Create a tabular array for the function [latex]g\left(x\correct)=\frac{1}{2}f\left(x\right)[/latex].

[latex]ten[/latex]	ii	four	6	8
[latex]f\left(10\right)[/latex]	1	3	7	11

Solution

The formula [latex]one thousand\left(x\right)=\frac{1}{2}f\left(x\right)[/latex] tells us that the output values of [latex]m[/latex] are one-half of the output values of [latex]f[/latex] with the same inputs. For example, we know that [latex]f\left(iv\right)=three[/latex]. Then

[latex]grand\left(four\right)=\frac{1}{2}f\left(4\correct)=\frac{1}{two}\left(three\right)=\frac{3}{2}[/latex]

Nosotros do the same for the other values to produce this table.

[latex]10[/latex]	[latex]2[/latex]	[latex]four[/latex]	[latex]6[/latex]	[latex]eight[/latex]
[latex]g\left(x\correct)[/latex]	[latex]\frac{1}{ii}[/latex]	[latex]\frac{3}{2}[/latex]	[latex]\frac{7}{2}[/latex]	[latex]\frac{11}{two}[/latex]

Try It 5

A function [latex]f[/latex] is given below. Create a tabular array for the function [latex]g\left(ten\right)=\frac{three}{4}f\left(10\correct)[/latex].

[latex]x[/latex]	2	iv	vi	viii
[latex]f\left(x\right)[/latex]	12	xvi	xx	0

Solution

Example 12: Recognizing a Vertical Stretch

Figure 17

The graph is a transformation of the toolkit function [latex]f\left(x\right)={x}^{3}[/latex]. Relate this new function [latex]k\left(ten\right)[/latex] to [latex]f\left(x\right)[/latex], and then find a formula for [latex]k\left(x\right)[/latex].

Solution

When trying to determine a vertical stretch or shift, it is helpful to look for a betoken on the graph that is relatively articulate. In this graph, it appears that [latex]g\left(ii\right)=2[/latex]. With the basic cubic function at the same input, [latex]f\left(2\right)={2}^{iii}=eight[/latex]. Based on that, information technology appears that the outputs of [latex]one thousand[/latex] are [latex]\frac{i}{4}[/latex] the outputs of the office [latex]f[/latex] considering [latex]g\left(two\right)=\frac{one}{4}f\left(2\right)[/latex]. From this we can adequately safely conclude that [latex]g\left(ten\right)=\frac{1}{iv}f\left(x\right)[/latex].

Nosotros can write a formula for [latex]g[/latex] by using the definition of the function [latex]f[/latex].

[latex]g\left(x\right)=\frac{i}{4}f\left(10\correct)=\frac{1}{4}{x}^{3}[/latex]

Endeavour Information technology half-dozen

Write the formula for the part that we get when nosotros stretch the identity toolkit role by a gene of 3, and and then shift it downward by ii units.

Solution

Horizontal Stretches and Compressions

Figure 18

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

Given a role [latex]y=f\left(x\correct)[/latex], the form [latex]y=f\left(bx\right)[/latex] results in a horizontal stretch or compression. Consider the role [latex]y={ten}^{two}[/latex]. The graph of [latex]y={\left(0.5x\correct)}^{ii}[/latex] is a horizontal stretch of the graph of the office [latex]y={x}^{two}[/latex] by a factor of 2. The graph of [latex]y={\left(2x\correct)}^{2}[/latex] is a horizontal pinch of the graph of the function [latex]y={10}^{2}[/latex] by a factor of 2.

A General Note: Horizontal Stretches and Compressions

Given a function [latex]f\left(10\right)[/latex], a new function [latex]g\left(x\right)=f\left(bx\correct)[/latex], where [latex]b[/latex] is a constant, is a horizontal stretch or horizontal compression of the part [latex]f\left(x\right)[/latex].

• If [latex]b>1[/latex], then the graph will be compressed by [latex]\frac{1}{b}[/latex].
• If [latex]0<b<1[/latex], then the graph will exist stretched by [latex]\frac{1}{b}[/latex].
• If [latex]b<0[/latex], then there volition be combination of a horizontal stretch or compression with a horizontal reflection.

How To: Given a clarification of a function, sketch a horizontal compression or stretch.

1. Write a formula to represent the role.
2. Fix [latex]yard\left(x\right)=f\left(bx\right)[/latex] where [latex]b>1[/latex] for a pinch or [latex]0<b<1[/latex]
   for a stretch.

Case xiii: 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, [latex]R[/latex], volition progress in 1 hour the aforementioned corporeality equally the original population does in 2 hours, and in ii hours, it volition progress equally much equally the original population does in four hours. Sketch a graph of this population.

Solution

Symbolically, we could write

[latex]\begin{cases}R\left(1\right)=P\left(ii\correct),\hfill \\ R\left(two\correct)=P\left(4\correct),\text{ and in general,}\hfill \\ R\left(t\correct)=P\left(2t\right).\hfill \end{cases}[/latex]

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

Effigy xix. (a) Original population graph (b) Compressed population graph

Example xiv: Finding a Horizontal Stretch for a Tabular Office

A part [latex]f\left(x\right)[/latex] is given below. Create a table for the function [latex]g\left(ten\right)=f\left(\frac{1}{two}x\right)[/latex].

[latex]x[/latex]	2	4	half-dozen	8
[latex]f\left(ten\correct)[/latex]	1	iii	7	eleven

Solution

The formula [latex]chiliad\left(x\right)=f\left(\frac{1}{ii}x\right)[/latex] tells us that the output values for [latex]one thousand[/latex] are the same as the output values for the function [latex]f[/latex] at an input one-half the size. Notice that we do not have enough data to decide [latex]yard\left(2\right)[/latex] considering [latex]thou\left(two\right)=f\left(\frac{ane}{2}\cdot two\right)=f\left(1\right)[/latex], and we do not have a value for [latex]f\left(1\right)[/latex] in our table. Our input values to [latex]one thousand[/latex] will need to be twice as big to go inputs for [latex]f[/latex] that nosotros can evaluate. For example, we can determine [latex]thou\left(4\right)\text{.}[/latex]

[latex]g\left(four\right)=f\left(\frac{i}{2}\cdot four\right)=f\left(two\correct)=1[/latex]

Nosotros do the aforementioned for the other values to produce the table below.

[latex]x[/latex]	4	8	12	16
[latex]g\left(x\right)[/latex]	i	3	vii	11

Figure 20

This figure shows the graphs of both of these sets of points.

Example fifteen: Recognizing a Horizontal Compression on a Graph

Chronicle the function [latex]chiliad\left(x\correct)[/latex] to [latex]f\left(ten\right)[/latex] in Figure 21.

Figure 21

Solution

The graph of [latex]k\left(10\right)[/latex] looks like the graph of [latex]f\left(x\right)[/latex] horizontally compressed. Because [latex]f\left(10\right)[/latex] ends at [latex]\left(6,4\right)[/latex] and [latex]k\left(10\right)[/latex] ends at [latex]\left(2,4\right)[/latex], we can come across that the [latex]x\text{-}[/latex] values have been compressed past [latex]\frac{i}{3}[/latex], considering [latex]vi\left(\frac{1}{3}\right)=2[/latex]. We might also notice that [latex]chiliad\left(two\correct)=f\left(half-dozen\right)[/latex] and [latex]grand\left(1\right)=f\left(3\right)[/latex]. Either way, we can describe this relationship as [latex]g\left(x\right)=f\left(3x\right)[/latex]. This is a horizontal compression by [latex]\frac{1}{3}[/latex].

Try It 7

Write a formula for the toolkit foursquare root function horizontally stretched past a factor of iii.

Solution

Combining Vertical and Horizontal Shifts

At present that we have ii transformations, we can combine them together. Vertical shifts are outside changes that touch the output ( [latex]y\text{-}[/latex] ) axis values and shift the function upwards or downward. Horizontal shifts are inside changes that affect the input ( [latex]ten\text{-}[/latex] ) axis values and shift the part left or correct. Combining the ii types of shifts volition crusade the graph of a office to shift up or downward and right or left.

How To: Given a role and both a vertical and a horizontal shift, sketch the graph.

1. Place 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. Employ the shifts to the graph in either society.

Example 16: Graphing Combined Vertical and Horizontal Shifts

Given [latex]f\left(x\right)=|10|[/latex], sketch a graph of [latex]h\left(x\correct)=f\left(ten+1\right)-3[/latex].

The function [latex]f[/latex] is our toolkit accented value function. We know that this graph has a V shape, with the point at the origin. The graph of [latex]h[/latex] has transformed [latex]f[/latex] in two ways: [latex]f\left(10+ane\right)[/latex] is a alter on the inside of the role, giving a horizontal shift left by 1, and the subtraction by 3 in [latex]f\left(ten+1\right)-3[/latex] is a change to the outside of the office, giving a vertical shift down by three. The transformation of the graph is illustrated in Figure 22.

Let us follow one bespeak of the graph of [latex]f\left(10\correct)=|x|[/latex].

• The point [latex]\left(0,0\right)[/latex] is transformed first by shifting left 1 unit: [latex]\left(0,0\right)\to \left(-ane,0\right)[/latex]
• The betoken [latex]\left(-1,0\correct)[/latex] is transformed side by side by shifting downwards 3 units: [latex]\left(-1,0\right)\to \left(-1,-3\right)[/latex]

Figure 22

Figure 23 is the graph of [latex]h[/latex].

Figure 23

Endeavour Information technology 8

Given [latex]f\left(x\correct)=|x|[/latex], sketch a graph of [latex]h\left(10\correct)=f\left(x - 2\right)+4[/latex].

Solution

Example 17: Identifying Combined Vertical and Horizontal Shifts

Write a formula for the graph shown in Figure 24, which is a transformation of the toolkit square root office.

Figure 24

Solution

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

[latex]h\left(x\correct)=f\left(x - i\correct)+ii[/latex]

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

[latex]h\left(x\right)=\sqrt{x - 1}+2[/latex]

Endeavour It nine

Write a formula for a transformation of the toolkit reciprocal function [latex]f\left(x\right)=\frac{1}{x}[/latex] that shifts the function's graph ane unit to the right and one unit upward.

Solution

Example xviii: Applying a Learning Model Equation

A common model for learning has an equation similar to [latex]k\left(t\right)=-{2}^{-t}+1[/latex], where [latex]thou[/latex] is the percentage of mastery that can be achieved after [latex]t[/latex] practice sessions. This is a transformation of the function [latex]f\left(t\right)={2}^{t}[/latex] shown in Figure 25. Sketch a graph of [latex]yard\left(t\correct)[/latex].

Effigy 25

Solution

This equation combines 3 transformations into one equation.

• A horizontal reflection: [latex]f\left(-t\right)={2}^{-t}[/latex]
• A vertical reflection: [latex]-f\left(-t\correct)=-{ii}^{-t}[/latex]
• A vertical shift: [latex]-f\left(-t\right)+1=-{ii}^{-t}+1[/latex]

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

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

This ways that the original points, (0,1) and (ane,two) get (0,0) and (1,1) after we use the transformations.

In Figure 26, the get-go graph results from a horizontal reflection. The second results from a vertical reflection. The third results from a vertical shift up 1 unit.

Effigy 26

Try It x

Given the toolkit function [latex]f\left(x\correct)={x}^{two}[/latex], graph [latex]thou\left(x\right)=-f\left(10\correct)[/latex] and [latex]h\left(x\correct)=f\left(-x\right)[/latex]. Take note of any surprising beliefs for these functions.

Solution

Performing a Sequence of Transformations

When combining transformations, it is very important to consider the order of the transformations. For case, vertically shifting by 3 so vertically stretching by 2 does not create the same graph as vertically stretching by 2 and so vertically shifting past three, because when we shift outset, both the original role and the shift get stretched, while just the original office gets stretched when nosotros stretch first.

When we meet an expression such as [latex]2f\left(x\correct)+3[/latex], which transformation should nosotros beginning with? The answer here follows nicely from the order of operations. Given the output value of [latex]f\left(x\right)[/latex], we first multiply past 2, causing the vertical stretch, then add iii, causing the vertical shift. In other words, multiplication earlier addition.

Horizontal transformations are a little trickier to think almost. When we write [latex]grand\left(10\right)=f\left(2x+3\correct)[/latex], for example, we have to remember well-nigh how the inputs to the function [latex]g[/latex] relate to the inputs to the function [latex]f[/latex]. Suppose nosotros know [latex]f\left(7\correct)=12[/latex]. What input to [latex]thousand[/latex] would produce that output? In other words, what value of [latex]x[/latex] volition allow [latex]one thousand\left(x\correct)=f\left(2x+3\right)=12?[/latex] We would need [latex]2x+iii=vii[/latex]. To solve for [latex]x[/latex], nosotros would starting time subtract iii, resulting in a horizontal shift, and then split by 2, causing a horizontal pinch.

This format ends upwards being very difficult to work with, because it is unremarkably much easier to horizontally stretch a graph before shifting. Nosotros can work around this by factoring inside the office.

[latex]f\left(bx+p\correct)=f\left(b\left(x+\frac{p}{b}\right)\right)[/latex]

Let's work through an example.

[latex]f\left(x\right)={\left(2x+four\right)}^{2}[/latex]

We tin can factor out a two.

[latex]f\left(x\right)={\left(ii\left(x+2\right)\correct)}^{2}[/latex]

Now nosotros can more conspicuously 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.

A General Note: Combining Transformations

When combining vertical transformations written in the form [latex]af\left(x\right)+thou[/latex], first vertically stretch by [latex]a[/latex] then vertically shift by [latex]k[/latex].

When combining horizontal transformations written in the form [latex]f\left(bx+h\right)[/latex], get-go horizontally shift past [latex]h[/latex] and so horizontally stretch past [latex]\frac{1}{b}[/latex].

When combining horizontal transformations written in the grade [latex]f\left(b\left(x+h\right)\right)[/latex], first horizontally stretch by [latex]\frac{ane}{b}[/latex] and then horizontally shift past [latex]h[/latex].

Horizontal and vertical transformations are independent. It does not thing whether horizontal or vertical transformations are performed start.

Example 19: Finding a Triple Transformation of a Tabular Function

Given the tabular array below for the function [latex]f\left(x\right)[/latex], create a tabular array of values for the office [latex]g\left(x\correct)=2f\left(3x\right)+1[/latex].

[latex]x[/latex]	6	12	18	24
[latex]f\left(x\correct)[/latex]	10	xiv	xv	17

Solution

At that place are three steps to this transformation, and we volition piece of work from the inside out. Starting with the horizontal transformations, [latex]f\left(3x\correct)[/latex] is a horizontal compression by [latex]\frac{i}{3}[/latex], which means we multiply each [latex]10\text{-}[/latex] value by [latex]\frac{i}{3}[/latex].

[latex]x[/latex]	2	four	6	8
[latex]f\left(3x\right)[/latex]	10	14	fifteen	17

Looking now to the vertical transformations, we start with the vertical stretch, which will multiply the output values by 2. Nosotros employ this to the previous transformation.

[latex]x[/latex]	2	4	6	8
[latex]2f\left(3x\correct)[/latex]	twenty	28	30	34

Finally, we tin can apply the vertical shift, which will add ane to all the output values.

[latex]10[/latex]	ii	4	6	8
[latex]1000\left(x\right)=2f\left(3x\right)+i[/latex]	21	29	31	35

Example xx: Finding a Triple Transformation of a Graph

Use the graph of [latex]f\left(x\right)[/latex] to sketch a graph of [latex]one thousand\left(x\correct)=f\left(\frac{1}{two}x+1\correct)-3[/latex].

Effigy 27

Solution

To simplify, allow's start by factoring out the within of the office.

[latex]f\left(\frac{1}{two}x+ane\right)-3=f\left(\frac{ane}{two}\left(10+2\correct)\right)-iii[/latex]

By factoring the inside, we can first horizontally stretch by 2, as indicated by the [latex]\frac{1}{2}[/latex] on the within of the office. 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 (four,0).

Figure 28

Next, we horizontally shift left past two units, as indicated by [latex]10+2[/latex].

Effigy 29

Final, nosotros vertically shift downward by 3 to complete our sketch, as indicated by the [latex]-iii[/latex] on the outside of the function.

Figure xxx

Primal Equations

Vertical shift	[latex]g\left(x\correct)=f\left(x\correct)+m[/latex] (up for [latex]grand>0[/latex] )
Horizontal shift	[latex]grand\left(ten\right)=f\left(x-h\correct)[/latex] (right for [latex]h>0[/latex] )
Vertical reflection	[latex]thou\left(x\correct)=-f\left(x\correct)[/latex]
Horizontal reflection	[latex]g\left(x\correct)=f\left(-x\right)[/latex]
Vertical stretch	[latex]chiliad\left(x\right)=af\left(x\correct)[/latex] ( [latex]a>0[/latex])
Vertical compression	[latex]1000\left(ten\right)=af\left(10\correct)[/latex] [latex]\left(0<a<i\right)[/latex]
Horizontal stretch	[latex]g\left(x\correct)=f\left(bx\right)[/latex] [latex]\left(0<b<ane\right)[/latex]
Horizontal compression	[latex]k\left(x\correct)=f\left(bx\right)[/latex] ( [latex]b>one[/latex] )

Key Concepts

• A role tin can be shifted vertically by adding a abiding to the output.
• A function tin exist shifted horizontally by adding a abiding to the input.
• Relating the shift to the context of a problem makes it possible to compare and interpret vertical and horizontal shifts.
• Vertical and horizontal shifts are oftentimes combined.
• A vertical reflection reflects a graph about the [latex]x\text{-}[/latex] axis. A graph can exist reflected vertically past multiplying the output by –1.
• A horizontal reflection reflects a graph most the [latex]y\text{-}[/latex] centrality. A graph tin can be reflected horizontally by multiplying the input by –i.
• A graph tin can be reflected both vertically and horizontally. The gild in which the reflections are applied does not affect the final graph.
• A function presented in tabular form can too be reflected past multiplying the values in the input and output rows or columns accordingly.
• A function presented every bit an equation can be reflected by applying transformations one at a time.
• Fifty-fifty functions are symmetric about the [latex]y\text{-}[/latex] axis, whereas odd functions are symmetric almost the origin.
• Even functions satisfy the status [latex]f\left(10\right)=f\left(-x\correct)[/latex].
• Odd functions satisfy the condition [latex]f\left(x\correct)=-f\left(-x\right)[/latex].
• A office can be odd, fifty-fifty, or neither.
• A function tin can exist compressed or stretched vertically by multiplying the output by a constant.
• A function can exist compressed or stretched horizontally past multiplying the input by a abiding.
• The order in which different transformations are applied does affect the final role. Both vertical and horizontal transformations must exist applied in the order given. All the same, a vertical transformation may exist combined with a horizontal transformation in whatever order.

Glossary

even function

a office whose graph is unchanged past horizontal reflection, [latex]f\left(x\correct)=f\left(-ten\right)[/latex], and is symmetric virtually the [latex]y\text{-}[/latex] axis

horizontal compression

a transformation that compresses a office'due south graph horizontally, past multiplying the input by a constant [latex]b>1[/latex]

horizontal reflection

a transformation that reflects a function'southward graph beyond the y-centrality by multiplying the input by [latex]-1[/latex]

horizontal shift

a transformation that shifts a function'south graph left or right past adding a positive or negative abiding to the input

horizontal stretch

a transformation that stretches a office's graph horizontally by multiplying the input by a constant [latex]0<b<ane[/latex]

odd function

a function whose graph is unchanged by combined horizontal and vertical reflection, [latex]f\left(x\correct)=-f\left(-x\right)[/latex], and is symmetric nearly the origin

vertical compression

a office transformation that compresses the office's graph vertically by multiplying the output past a constant [latex]0<a<1[/latex]

vertical reflection

a transformation that reflects a function's graph across the x-axis past multiplying the output by [latex]-1[/latex]

vertical shift

a transformation that shifts a office's graph up or down by adding a positive or negative constant to the output

vertical stretch

a transformation that stretches a function'southward graph vertically past multiplying the output by a abiding [latex]a>ane[/latex]

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

2. 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?

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

4. When examining the formula of a role that is the effect of multiple transformations, how tin can you tell a reflection with respect to the x-axis from a reflection with respect to the y-axis?

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

6. Write a formula for the function obtained when the graph of [latex]f\left(x\right)=\sqrt{x}[/latex] is shifted upwards 1 unit and to the left 2 units.

7. Write a formula for the office obtained when the graph of [latex]f\left(x\right)=|x|[/latex]
is shifted down 3 units and to the right ane unit.

8. Write a formula for the office obtained when the graph of [latex]f\left(x\right)=\frac{ane}{10}[/latex] is shifted down iv units and to the right iii units.

nine. Write a formula for the function obtained when the graph of [latex]f\left(x\right)=\frac{1}{{x}^{2}}[/latex] is shifted up ii units and to the left 4 units.

For the following exercises, draw how the graph of the function is a transformation of the graph of the original function [latex]f[/latex].

10. [latex]y=f\left(ten - 49\right)[/latex]

11. [latex]y=f\left(x+43\right)[/latex]

12. [latex]y=f\left(x+3\right)[/latex]

13. [latex]y=f\left(10 - 4\right)[/latex]

14. [latex]y=f\left(x\right)+5[/latex]

15. [latex]y=f\left(x\right)+8[/latex]

16. [latex]y=f\left(10\correct)-ii[/latex]

17. [latex]y=f\left(x\right)-7[/latex]

18. [latex]y=f\left(x - ii\right)+iii[/latex]

xix. [latex]y=f\left(10+4\right)-1[/latex]

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

20. [latex]f\left(x\right)=iv{\left(x+1\right)}^{2}-five[/latex]

21. [latex]g\left(10\right)=5{\left(ten+3\correct)}^{two}-2[/latex]

22. [latex]a\left(x\correct)=\sqrt{-ten+4}[/latex]

23. [latex]k\left(x\right)=-iii\sqrt{x}-1[/latex]

For the post-obit exercises, employ the graph of [latex]f\left(x\right)={2}^{x}[/latex] to sketch a graph of each transformation of [latex]f\left(x\right)[/latex].

24. [latex]g\left(x\right)={2}^{x}+ane[/latex]

25. [latex]h\left(x\right)={2}^{x}-iii[/latex]

26. [latex]w\left(x\correct)={2}^{x - 1}[/latex]

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

27. [latex]f\left(t\correct)={\left(t+one\right)}^{two}-three[/latex]

28. [latex]h\left(x\right)=|x - 1|+iv[/latex]

29. [latex]k\left(ten\correct)={\left(10 - 2\right)}^{3}-1[/latex]

thirty. [latex]m\left(t\right)=iii+\sqrt{t+two}[/latex]

31. Tabular representations for the functions [latex]f,k[/latex], and [latex]h[/latex] are given below. Write [latex]grand\left(x\right)[/latex] and [latex]h\left(x\correct)[/latex] as transformations of [latex]f\left(x\correct)[/latex].

[latex]x[/latex]	−2	−1	0	ane	ii
[latex]f\left(x\right)[/latex]	−two	−i	−3	1	ii

[latex]10[/latex]	−1	0	one	ii	3
[latex]one thousand\left(ten\right)[/latex]	−two	−1	−3	1	two

[latex]x[/latex]	−2	−1	0	1	2
[latex]h\left(x\correct)[/latex]	−1	0	−2	2	3

32. Tabular representations for the functions [latex]f,g[/latex], and [latex]h[/latex] are given below. Write [latex]thousand\left(x\right)[/latex] and [latex]h\left(x\right)[/latex] as transformations of [latex]f\left(ten\right)[/latex].

[latex]ten[/latex]	−2	−1	0	ane	2
[latex]f\left(x\right)[/latex]	−1	−three	4	2	1

[latex]x[/latex]	−3	−2	−one	0	1
[latex]m\left(ten\right)[/latex]	−1	−3	iv	two	1

[latex]x[/latex]	−2	−1	0	1	2
[latex]h\left(10\correct)[/latex]	−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.

33.

34.

35.

36.

37.

38.

39.

40.

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

41.

42.

For the post-obit exercises, utilise the graphs of the transformed toolkit functions to write a formula for each of the resulting functions.

43.

44.

45.

46.

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

47. [latex]f\left(x\right)=3{x}^{4}[/latex]

48. [latex]thou\left(10\right)=\sqrt{ten}[/latex]

49. [latex]h\left(ten\right)=\frac{1}{x}+3x[/latex]

50. [latex]f\left(x\correct)={\left(x - 2\right)}^{2}[/latex]

51. [latex]k\left(x\right)=2{ten}^{4}[/latex]

52. [latex]h\left(10\right)=2x-{x}^{iii}[/latex]

For the post-obit exercises, describe how the graph of each part is a transformation of the graph of the original function [latex]f[/latex].

53. [latex]k\left(x\right)=-f\left(x\right)[/latex]

54. [latex]g\left(ten\right)=f\left(-ten\right)[/latex]

55. [latex]one thousand\left(x\right)=4f\left(x\right)[/latex]

56. [latex]g\left(x\right)=6f\left(10\correct)[/latex]

57. [latex]g\left(x\right)=f\left(5x\right)[/latex]

58. [latex]g\left(x\right)=f\left(2x\right)[/latex]

59. [latex]g\left(x\right)=f\left(\frac{1}{three}ten\right)[/latex]

sixty. [latex]thousand\left(10\right)=f\left(\frac{1}{5}x\right)[/latex]

61. [latex]g\left(ten\correct)=3f\left(-10\right)[/latex]

62. [latex]g\left(x\right)=-f\left(3x\right)[/latex]

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

63. The graph of [latex]f\left(x\right)=|ten|[/latex] is reflected over the [latex]y[/latex] –axis and horizontally compressed by a cistron of [latex]\frac{ane}{4}[/latex] .

64. The graph of [latex]f\left(x\right)=\sqrt{x}[/latex] is reflected over the [latex]x[/latex] -axis and horizontally stretched by a cistron of 2.

65. The graph of [latex]f\left(x\right)=\frac{i}{{10}^{ii}}[/latex] is vertically compressed by a gene of [latex]\frac{1}{3}[/latex], and then shifted to the left 2 units and down 3 units.

66. The graph of [latex]f\left(x\correct)=\frac{1}{10}[/latex] is vertically stretched by a gene of viii, and so shifted to the right iv units and up two units.

67. The graph of [latex]f\left(x\right)={ten}^{ii}[/latex] is vertically compressed by a cistron of [latex]\frac{1}{2}[/latex], so shifted to the correct 5 units and up one unit of measurement.

68. The graph of [latex]f\left(x\right)={ten}^{2}[/latex] is horizontally stretched by a factor of 3, and so shifted to the left 4 units and downwardly iii units.

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

69. [latex]thou\left(10\right)=4{\left(x+1\right)}^{ii}-5[/latex]

70. [latex]g\left(x\correct)=five{\left(10+3\correct)}^{2}-2[/latex]

71. [latex]h\left(x\correct)=-2|x - 4|+3[/latex]

72. [latex]chiliad\left(x\correct)=-3\sqrt{x}-1[/latex]

73. [latex]k\left(10\right)=\frac{1}{2}{ten}^{iii}[/latex]

74. [latex]due north\left(x\right)=\frac{1}{3}|x - 2|[/latex]

75. [latex]p\left(10\right)={\left(\frac{1}{3}10\correct)}^{iii}-3[/latex]

76. [latex]q\left(ten\right)={\left(\frac{1}{4}ten\correct)}^{3}+1[/latex]

77. [latex]a\left(x\right)=\sqrt{-x+four}[/latex]

For the post-obit exercises, use the graph below to sketch the given transformations.

78. [latex]g\left(x\right)=f\left(x\right)-ii[/latex]

79. [latex]thousand\left(x\right)=-f\left(x\right)[/latex]

80. [latex]chiliad\left(10\right)=f\left(10+1\right)[/latex]

81. [latex]g\left(ten\right)=f\left(x - 2\right)[/latex]

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Source: https://courses.lumenlearning.com/precalcone/chapter/transformation-of-functions/

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