1.8 Transformations of Functions

Introduction

Translations of Functions

Vertical stretches and reflections

Combining shifts and stretches: why order sometimes matters

Summary

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

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

On the same axes as the plot of $y = r(x)$, sketch the following graphs: $y = g(x) = r(x) + 2$, $y = h(x) = r(x+1)$, and $y = f(x) = r(x+1) + 2$. Be sure to label the point on each of $g$, $h$, and $f$ that corresponds to $(2,-1)$ on the original graph of $r$. In addition, write one sentence to explain the overall transformations that have resulted in $g$, $h$, and $f$.

On the same axes as the plot of $y = s(x)$, sketch the following graphs: $y = k(x) = s(x) - 1$, $y = j(x) = s(x-2)$, and $y = m(x) = s(x-2) - 1$. Be sure to label the point on each of $k$, $j$, and $m$ that corresponds to $(-2,-3)$ on the original graph of $s$. In addition, write one sentence to explain the overall transformations that have resulted in $k$, $j$, and $m$.

Now consider the function $q(x) = x^2$. Determine a formula for the function that is given by $p(x) = q(x+3) - 4$. How is $p$ a transformation of $q$?

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

On the same axes as the plot of $y = r(x)$, sketch the following graphs: $y = g(x) = 3r(x)$ and $y = h(x) = \frac{1}{3}r(x)$. Be sure to label the point on $g$ and $h$ that corresponds to the point $(-2,1)$ on the original graph of $r$. In addition, write one sentence to explain the overall transformations that have resulted in $g$ and $h$ from $r$.

On the same axes as the plot of $y = s(x)$, sketch the following graphs: $y = k(x) = -s(x)$ and $y = j(x) = -\frac{1}{2}s(x)$. Be sure to label the point on $k$ and $j$ that corresponds to the point $(-2,-3)$ on the original graph of $s$. In addition, write one sentence to explain the overall transformations that have resulted in $k$ and $j$ from $s$.

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 $y = n(x) = \frac{1}{2}s(x-2)+2$. As above, be sure to label a key point on each graph that corresponds to a labeled point on the original parent function.

Describe in words how the function $y = m(x) = 2r(x+1)-1$ is the result of three elementary transformations of $y = r(x)$. Does the order in which these transformations occur matter? Why or why not?

Sketch an accurate graph of the transformation $y = p(x) = -\frac{1}{2}f(x-1)+2$. Write at least one sentence to explain how you developed the graph of $p$, and identify the point on $p$ that corresponds to the original point $(-2,2)$ on the graph of $f$.

Sketch an accurate graph of the transformation $y = q(x) = 2g(x+0.5)-0.75$. Write at least one sentence to explain how you developed the graph of $q$, and identify the point on $q$ that corresponds to the original point $(1.5,1.5)$ on the graph of $g$.

Is the function $y = r(x) = \frac{1}{2}(-f(x-1) - 4)$ the same function as $p$ in part (a) or different? Why? Explain in two different ways: discuss the algebraic similarities and differences between $p$ and $r$, and also discuss how each is a transformation of $f$.

Find a formula for a function $y = s(x)$ (in terms of $g$) that represents this transformation of $g$: 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 $g$.

Let $f(x) = x^2$.

• Let $g(x) = f(x) + 5$. Determine $AV_{[-3,-1]}$ and $AV_{[2,5]}$ for both $f$ and $g$. What do you observe? Why does this phenomenon occur? - Let $h(x) = f(x-2)$. For $f$, recall that you determined $AV_{[-3,-1]}$ and $AV_{[2,5]}$ in (a). In addition, determine $AV_{[-1,1]}$ and $AV_{[4,7]}$ for $h$. What do you observe? Why does this phenomenon occur? - Let $k(x) = 3f(x)$. Determine $AV_{[-3,-1]}$ and $AV_{[2,5]}$ for $k$, and compare the results to your earlier computations of $AV_{[-3,-1]}$ and $AV_{[2,5]}$ for $f$. What do you observe? Why does this phenomenon occur? - Finally, let $m(x) = 3f(x-2) + 5$. 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 $m$? Why? After making your conjecture, execute appropriate computations to see if your intuition is correct.

Consider the parent function $y = f(x) = x$.

• Consider the linear function in point-slope form given by $y = L(x) = -4(x-3) + 5$. What is the slope of this line? What is the most obvious point that lies on the line? - How can the function $L$ given in (a) be viewed as a transformation of the parent function $f$? Explain the roles of $3$, $-4$, and $5$, respectively. - 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 $f(x) = x$. Specifically discuss the transformation(s) involved. - 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.

We have explored the effects of adding a constant to the output of a function, $y = f(x) + a$, adding a constant to the input, $y = f(x+a)$, and multiplying the output of a function by a constant, $y = af(x)$. 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 $y = f(ax)$.

Let $f(x) = (x-2)^2 + 1$.

• Let $g(x) = f(4x)$, $h(x) = f(2x)$, $k(x) = f(0.5x)$, and $m(x) = f(0.25x)$. Use Desmos to plot these functions. Then, sketch and label $g$, $h$, $k$, and $m$ on the provided axes in Figure along with the graph of $f$. For each of the functions, label and identify its vertex, its $y$-intercept, and its $x$-intercepts. Axes for plotting $f$, $g$, $h$, $k$, and $m$ in part (a). Axes for plotting $f$, $r$, and $s$ from parts (c) and (d). - Based on your work in (a), how would you describe the effect(s) of the transformation $y = f(ax)$ where $a \gt 0$? What is the impact on the graph of $f$? Are any parts of the graph of $f$ unchanged? - Now consider the function $r(x) = f(-x)$. Observe that $r(-1) = f(1)$, $r(2) = f(-2)$, and so on. Without using a graphing utility, how do you expect the graph of $y = r(x)$ to compare to the graph of $y = f(x)$? Explain. Then test your conjecture by using a graphing utility and record the plots of $f$ and $r$ on the axes in Figure. - 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 $a \lt 0$?