Lesson 2 Shift and Stretch Solidify Understanding

Jump Start

Graph each function:

1.

$y = \frac{1}{2} {(x - 3)}^{2}$

2.

$y = - {(x + 2)}^{3} - 1$

3.

$y = - 3 + 2^{(x - 5)}$

Learning Focus

Transform the graph of $f (x) = \frac{1}{x}$ .

Write equations from graphs.

Predict the horizontal and vertical asymptotes of a function from the equation.

What other functions can be made from $f (x) = \frac{1}{x}$ ? What will their graphs look like?

Open Up the Math: Launch, Explore, Discuss

In the previous lesson, you were introduced to the function $f (x) = \frac{1}{x}$ . Before exploring the family of related functions, let’s clarify some of the features of $f (x) = \frac{1}{x}$ that can help with graphing.

1.

Use the graph of $f (x) = \frac{1}{x}$ to identify each of the following:

Horizontal Asymptote:

Vertical Asymptote:

Anchor Points:

$(1, \underset{―}{})$ and $(- 1, \underset{―}{})$

$(\frac{1}{2}, \underset{―}{})$ and $(- \frac{1}{2}, \underset{―}{})$

$(2, \underset{―}{})$ and $(- 2, \underset{―}{})$

Now you’re ready to use this information to figure out how the graph of $f (x) = \frac{1}{x}$ can be transformed. As you answer the problems that follow, look for patterns that you can generalize to describe the transformations of $f (x) = \frac{1}{x}$ .

In each of the following problems, you are given either a graph or a description of a function that is a transformation of $f (x) = \frac{1}{x}$ . Use your amazing math skills to find an equation for each.

2.

Equation:

3.

Equation:

4.

The function has a vertical asymptote at $x = - 3$ and a horizontal asymptote at $y = 0$ . It contains the points $(- 2, 1)$ and $(- 4, - 1)$ . The $y$ -intercept is $(0, \frac{1}{3})$ .

Equation:

5.

Equation:

6.

The function has a vertical asymptote at $x = 0$ and a horizontal asymptote at $y = 0$ . It contains the points $(1, 2)$ , $(- 1, - 2)$ , $(2, 1)$ , $(- 2, - 1)$ , $(\frac{1}{2}, 4)$ and $(\frac{- 1}{2}, - 4)$ .

Equation:

7.

Equation:

8.

Equation:

9.

The function has a vertical asymptote at $x = - 6$ and a horizontal asymptote at $y = - 3$ . It crosses the $x$ -axis at $- 6 \frac{1}{3}$ . It contains the points $(- 5, - 4)$ and $(- 7, - 2)$ .

Equation:

10.

___
$y = \frac{1}{x + b}$
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$y = b + \frac{1}{x}$
___
$y = \frac{b}{x}$
___
$y = \frac{- 1}{x}$
___
$y = \frac{1}{x - b}$

Reflection over the $x$ -axis.
Vertical shift of $b$ , making the horizontal asymptote $y = b$ .
Horizontal shift left $b$ , making the vertical asymptote $x = - b$ .
Vertical stretch by a factor of $b$ .
Horizontal shift right $b$ , making the vertical asymptote $x = b$ .

11.

Graph each of the following equations without using technology.

a.

$y = 1 - \frac{1}{x + 3}$

b.

$y = - 2 + \frac{3}{x - 4}$

12.

Describe the features of the function $f (x) = k + \frac{b}{x - h}$

Vertical asymptote:

Horizontal asymptote:

Vertical stretch factor:

Domain:

Range:

Ready for More?

You have already named the asymptotes and other features of $f (x) = k + \frac{b}{x - h}$ . Now, try naming the anchor points of the function.

Takeaways

Transformations of $f (x) = \frac{1}{x}$

Try one more: $f (x) = 3 - \frac{2}{x + 1}$

Lesson Summary

In this lesson, we learned to graph functions that are transformations of $f (x) = \frac{1}{x}$ . We learned that the transformations work just like other functions with horizontal shifts associated with the inputs to the function and the vertical effects associated with the outputs. Using these ideas, we also wrote equations to correspond with graphs and generalized each part of the equation in the form: $f (x) = k + \frac{b}{x - h}$ .

Retrieval

1.

Find the domain of $f (x) = \frac{1}{(x - 7) (x + 5) (x + 9)}$ .

2.

Find all of the roots of $g (x) = x^{3} + 6 x^{2} - x - 30$ , given that $x = - 5$ is a root of $g (x)$ . Then write $g (x)$ in factored form.

Roots:

Factored form: