# 3.2: Transformations of Common Graphs

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The transformation of graphs, using common functions, will be a skill that will bring insight to graphing functions quickly and painlessly. Anticipating how a graph of a function will look, and transforming old graphs to new graphs, is a skill we will explore in this section. Mastering this skill will give you a leg up on understanding analytic geometry, a key component to calculus.

Here are nine common functions and their graphs. Each of these functions will be referred to as “the parent function” for its family of transformed functions.

 $$f(x) = mx + b$$ $$f(x) = x^2$$ $$f(x) = |x|$$
 $$f(x) = \sqrt{x}$$ $$f(x) = x^3$$ $$f(x) = \ln x$$
 $$f(x) = \dfrac{1}{x}$$ $$f(x) = b^x$$ where $$b>1$$ $$f(x) = b^x$$ where $$0 < b< 1$$

Using the transformations described in Section $$3.1$$, the nine graphs shown on the previous page can be shifted, reflected, stretched, and compressed. The take-away for this section is to recognize the parent function from function notation and apply the appropriate transformations to quickly sketch a graph or visualize the graph in your head. No technology is necessary! It’s enough to have an understanding for functions and their shifting graphs.

##### Example 3.2.1

Name the parent function of the given function. Then describe in words the transformation(s) on the parent function.

1. $$y = |5x| − 6$$
2. $$y = 8 − (x + 9)^3$$
3. $$y = \dfrac{2}{x−10}$$

Solution

1. The parent function is $$y = |x|$$:

1. The parent function is $$y = x^3$$:

1. The parent function is $$y = \dfrac{1}{x}$$:

##### Example 3.2.2

Use the parent function and its transformation(s) to sketch the graph of the given function. It is much more important to create a rough but accurate sketch without technology than a detailed sketch with technology.

1. $$y = 2^x − 3$$
2. $$y = \ln(x + 2)$$

Solution

1. The parent function is $$y = b^x$$ where $$b = 2 > 1$$. Shift this graph, including its horizontal asymptote, down $$3$$ units.

1. The parent function is $$y = \ln x$$. Shift this graph, including its vertical asymptote, left $$2$$ units.

## Try It! (Exercises)

For exercises #1-18, use the parent functions (see graphs within this section), along with appropriate transformations, to create a sketch of the graph of the given function. Do not use graphing technology.

1. $$y = \dfrac{1}{x+5}$$
2. $$y = e^{x−4}$$
3. $$y = |x| − 6$$
4. $$y = −(x + 2)^3$$
5. $$y = 4 + x^2$$
6. $$y = \sqrt{x − 4}$$
7. $$y = \left( \dfrac{1}{2} \right)^x + 3$$
8. $$y = 1 − \sqrt{x}$$
9. $$y = |x − 5| + 2$$
10. $$y = 1 + (x − 6)^2$$
11. $$y = 4 − \dfrac{1}{x}$$
12. $$y = 3 + \ln (−x)$$
13. $$y = 2\sqrt{x} − 4$$
14. $$y = −2(x − 5)^2$$
15. $$y = 1 + x^3$$
16. $$y = −3|x − 7|$$
17. $$y = 5 + \sqrt{-x}$$
18. $$y= 3^{x+2} + 4$$
19. Use the graph of $$y = |x|$$ and transformations to discuss why the graph of the transformed function $$y = |-x|$$ is the same graph.
20. Use the graph of $$y = x^2$$ and transformations to discuss why the graph of the transformed function $$y = (-x)^2$$ is the same graph.
21. Using $$f(x) = x^2$$ as the parent function, write the function that would correspond to the transformations on $$f$$. No need to graph.

a. Reflected through the $$x$$-axis and shifted down $$12$$ units.

b. Shifted $$10$$ units left and $$25$$ units up.

1. Discuss two different approaches to transform the parent function $$y = \dfrac{1}{x}$$ to graph $$y = \dfrac{1}{2x}$$. What are the transformations? Discuss how and why each approach will produce the same graph for the new function $$y = \dfrac{1}{2x}$$.

This page titled 3.2: Transformations of Common Graphs is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Jennifer Freidenreich.