# 1.8: Transformations of Functions

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##### Motivating Questions
• How is the graph of $$y = g(x) = af(x-b) + c$$ related to the graph of $$y = f(x)\text{?}$$
• What do we mean by “transformations” of a given function $$f\text{?}$$ How are translations and vertical stretches of a function examples of transformations?

In our preparation for calculus, we aspire to understand functions from a wide range of perspectives and to become familiar with a library of basic functions. So far, two basic families functions we have considered are linear functions and quadratic functions, the simplest of which are $$L(x) = x$$ and $$Q(x) = x^2\text{.}$$ As we progress further, we will endeavor to understand a “parent” function as the most fundamental member of a family of functions, as well as how other similar but more complicated functions are the result of transforming the parent function.

Informally, a transformation of a given function is an algebraic process by which we change the function to a related function that has the same fundamental shape, but may be shifted, reflected, and/or stretched in a systematic way. For example, among all quadratic functions, the simplest is the parent function $$Q(x) = x^2\text{,}$$ but any other quadratic function such as $$g(x) = -3(x-5)^2 + 4$$ can also be understood in relation to the parent function. We say that “$$g$$ is a transformation of $$f\text{.}$$”

In Preview Activity $$\PageIndex{1}$$, we investigate the effects of the constants $$a\text{,}$$ $$b\text{,}$$ and $$c$$ in generating the function $$g(x) = af(x-b) + c$$ in the context of already knowing the function $$f\text{.}$$

##### Preview Activity $$\PageIndex{1}$$

Open a new Desmos graph and define the function $$f(x) = x^2\text{.}$$ Adjust the window so that the range is for $$-4 \le x \le 4$$ and $$-10 \le y \le 10\text{.}$$

1. In Desmos, define the function $$g(x) = f(x) + a\text{.}$$ (That is, in Desmos on line 2, enter g(x) = f(x) + a.) You will get prompted to add a slider for $$a\text{.}$$ Do so.

Explore by moving the slider for $$a$$ and write at least one sentence to describe the effect that changing the value of $$a$$ has on the graph of $$g\text{.}$$

2. Next, define the function $$h(x) = f(x-b)\text{.}$$ (That is, in Desmos on line 4, enter h(x) = f(x-b) and add the slider for $$b\text{.}$$)

Move the slider for $$b$$ and write at least one sentence to describe the effect that changing the value of $$b$$ has on the graph of $$h\text{.}$$

3. Now define the function $$p(x) = cf(x)\text{.}$$ (That is, in Desmos on line 6, enter p(x) = cf(x) and add the slider for $$c\text{.}$$)

Move the slider for $$c$$ and write at least one sentence to describe the effect that changing the value of $$c$$ has on the graph of $$p\text{.}$$ In particular, when $$c = -1\text{,}$$ how is the graph of $$p$$ related to the graph of $$f\text{?}$$

4. Finally, click on the icons next to $$g\text{,}$$ $$h\text{,}$$ and $$p$$ to temporarily hide them, and go back to Line 1 and change your formula for $$f\text{.}$$ You can make it whatever you'd like, but try something like $$f(x) = x^2 + 2x + 3$$ or $$f(x) = x^3 - 1\text{.}$$ Then, investigate with the sliders $$a\text{,}$$ $$b\text{,}$$ and $$c$$ to see the effects on $$g\text{,}$$ $$h\text{,}$$ and $$p$$ (unhiding them appropriately). Write a couple of sentences to describe your observations of your explorations.

## Translations of Functions

We begin by summarizing two of our findings in Preview Activity $$\PageIndex{1}$$.

##### Vertical Translation of a Function.

Given a function $$y = f(x)$$ and a real number $$a\text{,}$$ the transformed function $$y = g(x) = f(x) + a$$ is a vertical translation of the graph of $$f\text{.}$$ That is, every point $$(x,f(x))$$ on the graph of $$f$$ gets shifted vertically to the corresponding point $$(x,f(x)+a)$$ on the graph of $$g\text{.}$$

As we found in our Desmos explorations in the preview activity, is especially helpful to see the effects of vertical translation dynamically.

Move the slider 1  by clicking and dragging on the red point to see how changing $$a$$ affects the graph of $$y = f(x) + a\text{,}$$ which appears in blue. The graph of $$y = f(x)$$ will appear in grey and remain fixed.

In a vertical translation, the graph of $$g$$ lies above the graph of $$f$$ whenever $$a \gt 0\text{,}$$ while the graph of $$g$$ lies below the graph of $$f$$ whenever $$a \lt 0\text{.}$$ In Figure $$\PageIndex{2}$$, we see the original parent function $$f(x) = |x|$$ along with the resulting transformation $$g(x) = f(x)-3\text{,}$$ which is a downward vertical shift of $$3$$ units. Note particularly that every point on the original graph of $$f$$ is moved $$3$$ units down; we often indicate this by an arrow and labeling at least one key point on each graph. Figure $$\PageIndex{2}$$ A vertical translation, $$g\text{,}$$ of the function $$y = f(x) = |x|\text{.}$$ Figure $$\PageIndex{3}$$ A horizontal translation, $$h\text{,}$$ of a different function $$y = f(x)\text{.}$$

In Figure $$\PageIndex{3}$$, we see a horizontal translation of the original function $$f$$ that shifts its graph $$2$$ units to the right to form the function $$h\text{.}$$ Observe that $$f$$ is not a familiar basic function; transformations may be applied to any original function we desire.

From an algebraic point of view, horizontal translations are slightly more complicated than vertical ones. Given $$y = f(x)\text{,}$$ if we define the transformed function $$y = h(x) = f(x-b)\text{,}$$ observe that

$h(x+b) = f( (x+b) - b ) = f(x)\text{.} \nonumber$

This shows that for an input of $$x+b$$ in $$h\text{,}$$ the output of $$h$$ is the same as the output of $$f$$ that corresponds to an input of simply $$x\text{.}$$ Hence, in Figure $$\PageIndex{3}$$, the formula for $$h$$ in terms of $$f$$ is $$h(x) = f(x-2)\text{,}$$ since an input of $$x+2$$ in $$h$$ will result in the same output as an input of $$x$$ in $$f\text{.}$$ For example, $$h(2) = f(0)\text{,}$$ which aligns with the graph of $$h$$ being a shift of the graph of $$f$$ to the right by $$2$$ units.

Again, it's instructive to see the effects of horizontal translation dynamically.

Move the slider by clicking and dragging on the red point to see how changing $$b$$ affects the graph of $$y = f(x-b)\text{,}$$ which appears in blue. The graph of $$y = f(x)$$ will appear in grey and remain fixed.

Overall, we have the following general principle.

##### Horizontal Translation of a Function.

Given a function $$y = f(x)$$ and a real number $$b\text{,}$$ the transformed function $$y = h(x) = f(x-b)$$ is a horizontal translation of the graph of $$f\text{.}$$ That is, every point $$(x,f(x))$$ on the graph of $$f$$ gets shifted horizontally to the corresponding point $$(x+b,f(x))$$ on the graph of $$g\text{.}$$

We emphasize that in the horizontal translation $$h(x) = f(x-b)\text{,}$$ if $$b \gt 0$$ the graph of $$h$$ lies $$b$$ units to the right of $$f\text{,}$$ while if $$b \lt 0\text{,}$$ $$h$$ lies $$b$$ units to the left of $$f\text{.}$$

##### Activity $$\PageIndex{2}$$

Consider the functions $$r$$ and $$s$$ given in Figure $$\PageIndex{5}$$ and Figure $$\PageIndex{6}$$ Figure $$\PageIndex{5}$$ A parent function $$r\text{.}$$ Figure $$\PageIndex{6}$$ A parent function $$s\text{.}$$
1. On the same axes as the plot of $$y = r(x)\text{,}$$ sketch the following graphs: $$y = g(x) = r(x) + 2\text{,}$$ $$y = h(x) = r(x+1)\text{,}$$ and $$y = f(x) = r(x+1) + 2\text{.}$$ Be sure to label the point on each of $$g\text{,}$$ $$h\text{,}$$ and $$f$$ that corresponds to $$(-2,-1)$$ on the original graph of $$r\text{.}$$ In addition, write one sentence to explain the overall transformations that have resulted in $$g\text{,}$$ $$h\text{,}$$ and $$f\text{.}$$
2. On the same axes as the plot of $$y = s(x)\text{,}$$ sketch the following graphs: $$y = k(x) = s(x) - 1\text{,}$$ $$y = j(x) = s(x-2)\text{,}$$ and $$y = m(x) = s(x-2) - 1\text{.}$$ Be sure to label the point on each of $$k\text{,}$$ $$j\text{,}$$ and $$m$$ that corresponds to $$(-2,-3)$$ on the original graph of $$r\text{.}$$ In addition, write one sentence to explain the overall transformations that have resulted in $$k\text{,}$$ $$j\text{,}$$ and $$m\text{.}$$
3. Now consider the function $$q(x) = x^2\text{.}$$ Determine a formula for the function that is given by $$p(x) = q(x+3) - 4\text{.}$$ How is $$p$$ a transformation of $$q\text{?}$$

## Vertical stretches and reflections

So far, we have seen the possible effects of adding a constant value to function output —$$f(x)+a$$— and adding a constant value to function input — $$f(x+b)\text{.}$$ Each of these actions results in a translation of the function's graph (either vertically or horizontally), but otherwise leaving the graph the same. Next, we investigate the effects of multiplication the function's output by a constant.

##### Example $$\PageIndex{7}$$

Given the parent function $$y = f(x)$$ pictured in Figure $$\PageIndex{8}$$, what are the effects of the transformation $$y = v(x) = cf(x)$$ for various values of $$c\text{?}$$

Solution

We first investigate the effects of $$c = 2$$ and $$c = \frac{1}{2}\text{.}$$ For $$v(x) = 2f(x)\text{,}$$ the algebraic impact of this transformation is that every output of $$f$$ is multiplied by $$2\text{.}$$ This means that the only output that is unchanged is when $$f(x) = 0\text{,}$$ while any other point on the graph of the original function $$f$$ will be stretched vertically away from the $$x$$-axis by a factor of $$2\text{.}$$ We can see this in Figure $$\PageIndex{8}$$ where each point on the original dark blue graph is transformed to a corresponding point whose $$y$$-coordinate is twice as large, as partially indicated by the red arrows. Figure $$\PageIndex{8}$$ The parent function $$y = f(x)$$ along with two different vertical stretches, $$v$$ and $$u\text{.}$$ Figure $$\PageIndex{9}$$ The parent function $$y = f(x)$$ along with a vertical reflection, $$z\text{,}$$ and a corresponding stretch, $$w\text{.}$$

In contrast, the transformation $$u(x) = \frac{1}{2}f(x)$$ is stretched vertically by a factor of $$\frac{1}{2}\text{,}$$ which has the effect of compressing the graph of $$f$$ towards the $$x$$-axis, as all function outputs of $$f$$ are multiplied by $$\frac{1}{2}\text{.}$$ For instance, the point $$(0,-2)$$ on the graph of $$f$$ is transformed to the graph of $$(0,-1)$$ on the graph of $$u\text{,}$$ and others are transformed as indicated by the purple arrows.

To consider the situation where $$c \lt 0\text{,}$$ we first consider the simplest case where $$c = -1$$ in the transformation $$z(x) = -f(x)\text{.}$$ Here the impact of the transformation is to multiply every output of the parent function $$f$$ by $$-1\text{;}$$ this takes any point of form $$(x,y)$$ and transforms it to $$(x,-y)\text{,}$$ which means we are reflecting each point on the original function's graph across the $$x$$-axis to generate the resulting function's graph. This is demonstrated in Figure $$\PageIndex{9}$$ where $$y = z(x)$$ is the reflection of $$y = f(x)$$ across the $$x$$-axis.

Finally, we also investigate the case where $$c = -2\text{,}$$ which generates $$y = w(x) = -2f(x)\text{.}$$ Here we can think of $$-2$$ as $$-2 = 2(-1)\text{:}$$ the effect of multiplying by $$-1$$ first reflects the graph of $$f$$ across the $$x$$-axis (resulting in $$w$$), and then multiplying by $$2$$ stretches the graph of $$z$$ vertically to result in $$w\text{,}$$ as shown in Figure $$\PageIndex{9}$$

As with vertical and horizontal translation, it's particularly instructive to see the effects of vertical scaling in a dynamic way.

Move the slider by clicking and dragging on the red point to see how changing $$c$$ affects the graph of $$y = cf(x)\text{,}$$ which is shown in blue. The graph of $$y = f(x)$$ will appear in grey and remain fixed.

We summarize and generalize our observations from Example $$\PageIndex{7}$$ and Figure $$\PageIndex{10}$$ as follows.

##### Vertical Scaling of a Function.

Given a function $$y = f(x)$$ and a real number $$c \gt 0\text{,}$$ the transformed function $$y = v(x) = cf(x)$$ is a vertical stretch of the graph of $$f\text{.}$$ Every point $$(x,f(x))$$ on the graph of $$f$$ gets stretched vertically to the corresponding point $$(x,cf(x))$$ on the graph of $$v\text{.}$$ If $$0 \lt c \lt 1\text{,}$$ the graph of $$v$$ is a compression of $$f$$ toward the $$x$$-axis; if $$c \gt 1\text{,}$$ the graph of $$v$$ is a stretch of $$f$$ away from the $$x$$-axis. Points where $$f(x) = 0$$ are unchanged by the transformation.

Given a function $$y = f(x)$$ and a real number $$c \lt 0\text{,}$$ the transformed function $$y = v(x) = cf(x)$$ is a reflection of the graph of $$f$$ across the $$x$$-axis followed by a vertical stretch by a factor of $$|c|\text{.}$$

##### Exercise $$\PageIndex{1}$$ Figure $$\PageIndex{11}$$ A parent function $$r\text{.}$$ Figure $$\PageIndex{12}$$ A parent function $$s\text{.}$$
1. On the same axes as the plot of $$y = r(x)\text{,}$$ sketch the following graphs: $$y = g(x) = 3r(x)$$ and $$y = h(x) = \frac{1}{3}r(x)\text{.}$$ Be sure to label several points on each of $$r\text{,}$$ $$g\text{,}$$ and $$h$$ with arrows to indicate their correspondence. In addition, write one sentence to explain the overall transformations that have resulted in $$g$$ and $$h$$ from $$r\text{.}$$
2. On the same axes as the plot of $$y = s(x)\text{,}$$ sketch the following graphs: $$y = k(x) = -s(x)$$ and $$y = j(x) = -\frac{1}{2}s(x)\text{.}$$ Be sure to label several points on each of $$s\text{,}$$ $$k\text{,}$$ and $$j$$ with arrows to indicate their correspondence. In addition, write one sentence to explain the overall transformations that have resulted in $$k$$ and $$j$$ from $$s\text{.}$$
3. On the additional copies of the two figures below, sketch the graphs of the following transformed functions: $$y = m(x) = 2r(x+1)-1$$ (at left) and $$y = n(x) = \frac{1}{2}s(x-2)+2\text{.}$$ As above, be sure to label several points on each graph and indicate their correspondence to points on the original parent function.
4. Describe in words how the function $$y = m(x) = 2r(x+1)-1$$ is the result of three elementary transformations of $$y = r(x)\text{.}$$ Does the order in which these transformations occur matter? Why or why not?

## Combining shifts and stretches: why order sometimes matters

In the final question of Activity $$\PageIndex{3}$$, we considered the transformation $$y = m(x) = 2r(x+1)-1$$ of the original function $$r\text{.}$$ There are three different basic transformations involved: a vertical shift of $$1$$ unit down, a horizontal shift of $$1$$ unit left, and a vertical stretch by a factor of $$2\text{.}$$ To understand the order in which these transformations are applied, it's essential to remember that a function is a process that converts inputs to outputs.

By the algebraic rule for $$m\text{,}$$ $$m(x) = 2r(x+1)-1\text{.}$$ In words, this means that given an input $$x$$ for $$m\text{,}$$ we do the following processes in this particular order:

1. add $$1$$ to $$x$$ and then apply the function $$r$$ to the quantity $$x+1\text{;}$$
2. multiply the output of $$r(x+1)$$ by $$2\text{;}$$
3. subtract $$1$$ from the output of $$2r(x+1)\text{.}$$

These three steps correspond to three basic transformations: (1) shift the graph of $$r$$ to the left by $$1$$ unit; (2) stretch the resulting graph vertically by a factor of $$2\text{;}$$ (3) shift the resulting graph vertically by $$-1$$ units. We can see the graphical impact of these algebraic steps by taking them one at a time. In Figure $$\PageIndex{14}$$, we see the function $$p$$ that results from a shift $$1$$ unit left of the parent function in Figure $$\PageIndex{13}$$ (Each time we take an additional step, we will de-emphasize the preceding function by having it appear in lighter color and dashed.) Figure $$\PageIndex{13}$$ The parent function $$y = r(x)\text{.}$$ Figure $$\PageIndex{14}$$ The parent function $$y = r(x)$$ along with the horizontal shift $$y = p(x) = r(x+1)\text{.}$$

Continuing, we now consider the function $$q(x) = 2p(x) = 2r(x+1)\text{,}$$ which results in a vertical stretch of $$p$$ away from the $$x$$-axis by a factor of $$2\text{,}$$ as seen in Figure $$\PageIndex{15}$$ Figure $$\PageIndex{15}$$ The function $$y = q(x) = 2p(x) = 2r(x+1)$$ along with graphs of $$p$$ and $$r\text{.}$$ Figure $$\PageIndex{16}$$ The function $$y = m(x) = q(x)-1 = 2r(x+1) - 1$$ along with graphs of $$q\text{,}$$ $$p$$ and $$r\text{.}$$

Finally, we arrive at $$y = m(x) = 2r(x+1) - 1$$ by subtracting $$1$$ from $$q(x) = 2r(x+1)\text{;}$$ this of course is a vertical shift of $$-1$$ units, and produces the graph of $$m$$ shown in red in Figure 1.8.16. We can also track the point $$(2,-1)$$ on the original parent function: it first moves left $$1$$ unit to $$(1,-1)\text{,}$$ then it is stretched vertically by a factor of $$2$$ away from the $$x$$-axis to $$(1,-2)\text{,}$$ and lastly is shifted $$1$$ unit down to the point $$(1,-3)\text{,}$$ which we see on the graph of $$m\text{.}$$

While there are some transformations that can be executed in either order (such as a combination of a horizontal translation and a vertical translation, as seen in part (b) of Activity $$\PageIndex{2}$$), in other situations order matters. For instance, in our preceding discussion, we have to apply the vertical stretch before applying the vertical shift. Algebraically, this is because

$2r(x+1) - 1 \ne 2[r(x+1)-1]\text{.} \nonumber$

The quantity $$2r(x+1) - 1$$ multiplies the function $$r(x+1)$$ by $$2$$ first (the stretch) and then the vertical shift follows; the quantity $$2[r(x+1) - 1]$$ shifts the function $$r(x+1)$$ down $$1$$ unit first, and then executes a vertical stretch by a factor of $$2\text{.}$$ In the latter scenario, the point $$(1,-1)$$ that lies on $$r(x+1)$$ gets transformed first to $$(1,-2)$$ and then to $$(1,-4)\text{,}$$ which is not the same as the point $$(1,-3)$$ that lies on $$m(x) = 2r(x+1) - 1\text{.}$$

##### Activity $$\PageIndex{4}$$

Consider the functions $$f$$ and $$g$$ given in Figure $$\PageIndex{17}$$ and Figure $$\PageIndex{18}$$. Figure $$\PageIndex{17}$$ A parent function $$f\text{.}$$ Figure $$\PageIndex{18}$$ A parent function $$g\text{.}$$
1. Sketch an accurate graph of the transformation $$y = p(x) = -\frac{1}{2}f(x-1)+2\text{.}$$ Write at least one sentence to explain how you developed the graph of $$p\text{,}$$ and identify the point on $$p$$ that corresponds to the original point $$(-2,2)$$ on the graph of $$f\text{.}$$
2. Sketch an accurate graph of the transformation $$y = q(x) = 2g(x+0.5)-0.75\text{.}$$ Write at least one sentence to explain how you developed the graph of $$p\text{,}$$ and identify the point on $$q$$ that corresponds to the original point $$(1.5,1.5)$$ on the graph of $$g\text{.}$$
3. Is the function $$y = r(x) = \frac{1}{2}(-f(x-1) - 4)$$ the same function as $$p$$ or different? Why? Explain in two different ways: discuss the algebraic similarities and differences between $$p$$ and $$r\text{,}$$ and also discuss how each is a transformation of $$f\text{.}$$
4. Find a formula for a function $$y = s(x)$$ (in terms of $$g$$) that represents this transformation of $$g\text{:}$$ a horizontal shift of $$1.25$$ units left, followed by a reflection across the $$x$$-axis and a vertical stretch by a factor of $$2.5$$ units, followed by a vertical shift of $$1.75$$ units. Sketch an accurate, labeled graph of $$s$$ on the following axes along with the given parent function $$g\text{.}$$

## Summary

• The graph of $$y = g(x) = af(x-b) + c$$ is related to the graph of $$y = f(x)$$ by a sequence of transformations. First, there is horizontal shift of $$|b|$$ units to the right ($$b \gt 0$$) or left ($$b \lt 0$$). Next, there is a vertical stretch by a factor of $$|a|$$ (along with a reflection across $$y = 0$$ in the case where $$a \lt 0$$). Finally, there's a vertical shift of $$c$$ units.
• A transformation of a given function $$f$$ is a process by which the graph may be shifted or stretched to generate a new, related function with fundamentally the same shape. In this section we considered four different ways this can occur: through a horizontal translation (shift), through a reflection across the line $$y = 0$$ (the $$x$$-axis), through a vertical scaling (stretch) that multiplies every output of a function by the same constant, and through a vertical translation (shift). Each of these individual processes is itself a transformation, and they may be combined in various ways to create more complicated transformations.

## Exercises

##### 6.

Let $$f(x) = x^2\text{.}$$

1. Let $$g(x) = f(x) + 5\text{.}$$ Determine $$AV_{[-3,-1]}$$ and $$AV_{[2,5]}$$ for both $$f$$ and $$g\text{.}$$ What do you observe? Why does this phenomenon occur?
2. Let $$h(x) = f(x-2)\text{.}$$ For $$f\text{,}$$ recall that you determined $$AV_{[-3,-1]}$$ and $$AV_{[2,5]}$$ in (a). In addition, determine $$AV_{[-1,1]}$$ and $$AV_{[4,7]}$$ for $$h\text{.}$$ What do you observe? Why does this phenomenon occur?
3. Let $$k(x) = 3f(x)\text{.}$$ Determine $$AV_{[-3,-1]}$$ and $$AV_{[2,5]}$$ for $$k\text{,}$$ and compare the results to your earlier computations of $$AV_{[-3,-1]}$$ and $$AV_{[2,5]}$$ for $$f\text{.}$$ What do you observe? Why does this phenomenon occur?
4. Finally, let $$m(x) = 3f(x-2) + 5\text{.}$$ Without doing any computations, what do you think will be true about the relationship between $$AV_{[-3,-1]}$$ for $$f$$ and $$AV_{[-1,1]}$$ for $$m\text{?}$$ Why? After making your conjecture, execute appropriate computations to see if your intuition is correct.
##### 7.

Consider the parent function $$y = f(x) = x\text{.}$$

1. Consider the linear function in point-slope form given by $$y = L(x) = -4(x-3) + 5\text{.}$$ What is the slope of this line? What is the most obvious point that lies on the line?
2. How can the function $$L$$ given in (a) be viewed as a transformation of the parent function $$f\text{?}$$ Explain the roles of $$3\text{,}$$ $$-4\text{,}$$ and $$5\text{,}$$ respectively.
3. Explain why any non-vertical line of the form $$P(x) = m(x-x_0) + y_0$$ can be thought of as a transformation of the parent function $$f(x) = x\text{.}$$ Specifically discuss the transformation(s) involved.
4. Find a formula for the transformation of $$f(x) = x$$ that corresponds to a horizontal shift of $$7$$ units left, a reflection across $$y = 0$$ and vertical stretch of $$3$$ units away from the $$x$$-axis, and a vertical shift of $$-11$$ units.
##### 8.

We have explored the effects of adding a constant to the output of a function, $$y = f(x) + a\text{,}$$ adding a constant to the input, $$y = f(x+a)\text{,}$$ and multiplying the output of a function by a constant, $$y = af(x)\text{.}$$ There is one remaining natural transformation to explore: multiplying the input to a function by a constant. In this exercise, we consider the effects of the constant $$a$$ in transforming a parent function $$f$$ by the rule $$y = f(ax)\text{.}$$

Let $$f(x) = (x-2)^2 + 1\text{.}$$

1. Let $$g(x) = f(4x)\text{,}$$ $$h(x) = f(2x)\text{,}$$ $$k(x) = f(0.5x)\text{,}$$ and $$m(x) = f(0.25x)\text{.}$$ Use Desmos to plot these functions. Then, sketch and label $$g\text{,}$$ $$h\text{,}$$ $$k\text{,}$$ and $$m$$ on the provided axes in Figure $$\PageIndex{19}$$ along with the graph of $$f\text{.}$$ For each of the functions, label and identify its vertex, its $$y$$-intercept, and its $$x$$-intercepts. Figure $$\PageIndex{19}$$ Axes for plotting $$f\text{,}$$ $$g\text{,}$$ $$h\text{,}$$ $$k\text{,}$$ and $$m$$ in part (a). Figure $$\PageIndex{20}$$ Axes for plotting $$f\text{,}$$ $$r\text{,}$$ and $$s$$ from parts (c) and (d).

2. Based on your work in (a), how would you describe the effect(s) of the transformation $$y = f(ax)$$ where $$a \gt 0\text{?}$$ What is the impact on the graph of $$f\text{?}$$ Are any parts of the graph of $$f$$ unchanged?

3. Now consider the function $$r(x) = f(-x)\text{.}$$ Observe that $$r(-1) = f(1)\text{,}$$ $$r(2) = f(-2)\text{,}$$ and so on. Without using a graphing utility, how do you expect the graph of $$y = r(x)$$ to compare to the graph of $$y = f(x)\text{?}$$ Explain. Then test your conjecture by using a graphing utility and record the plots of $$f$$ and $$r$$ on the axes in Figure $$\PageIndex{20}$$

4. How do you expect the graph of $$s(x) = f(-2x)$$ to appear? Why? More generally, how does the graph of $$y = f(ax)$$ compare to the graph of $$y = f(x)$$ in the situation where $$a \lt 0\text{?}$$

<1> Huge thanks to the amazing David Austin for making these interactive javascript graphics for the text.

This page titled 1.8: Transformations of Functions is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Matthew Boelkins, David Austin & Steven Schlicker (ScholarWorks @Grand Valley State University) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.