Lesson 2E Shift and Stretch Solidify Understanding

Jump Start

Graph each function:

1.

a blank 17 by 17 grid

2.

a blank 17 by 17 grid

3.

a blank 17 by 17 grid

Learning Focus

Transform the graph of .

Write equations from graphs.

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

What other functions can be made from ? What will their graphs look like?

Open Up the Math: Launch, Explore, Discuss

In the previous lesson, you were introduced to the function . Before exploring the family of related functions, let’s clarify some of the features of that can help with graphing.

1.

Use the graph of to identify each of the following:

a curved line in the lower left quadrant and a curved line in the top right quadrant both with vertical and horizontal asymptotes at 0 and points at (-1,-1) and (1,1) representing f of x = 1 over x x–10–10–10–5–5–5555101010y–10–10–10–5–5–5555101010000

Horizontal Asymptote:

Vertical Asymptote:

Anchor Points:

and

and

and

Now you’re ready to use this information to figure out how the graph of can be transformed. As you answer the problems that follow, look for patterns that you can generalize to describe the transformations of .

In each of the following problems, you are given either a graph or a description of a function that is a transformation of . Use your amazing math skills to find an equation for each.

2.

the above graph translated up 5 units representing a transformation of the function f of x = 1 over x. there are now points at (-1,4) and (1,6) and a vertical asymptote at 0 and a horizontal asymptote at 5 x–10–10–10–5–5–5555101010y–5–5–5555101010000

Equation:

3.

the function f of x = 1 over x is graphed on a coordinate plane and reflected over either the x or y axis x–10–10–10–5–5–5555101010y–10–10–10–5–5–5555101010000

Equation:

4.

The function has a vertical asymptote at and a horizontal asymptote at . It contains the points and . The -intercept is .

Equation:

5.

the function f of x = 1 over x is graphed and translated 2 units to the left creating a vertical asymptote at 2 x–5–5–5555101010y–5–5–5555000

Equation:

6.

The function has a vertical asymptote at and a horizontal asymptote at . It contains the points , , , , and .

Equation:

7.

the function f of x = 1 over x is graphed and translated 1 unit to the right and 4 units down creating a vertical asymptote at -1 and a horizontal asymptote at -4 x–5–5–5555101010y–10–10–10–5–5–5555000

Equation:

8.

the function f of x = 1 over x is graphed and translated 5 units to the right and 2 units up creating a vertical asymptote at 5 and a horizontal asymptote at 2 x–5–5–5555101010y–5–5–5555101010000

Equation:

9.

The function has a vertical asymptote at and a horizontal asymptote at . It crosses the -axis at . It contains the points and .

Equation:

10.

Match each equation to the phrase that describes the transformation from .

  1. ___

  2. ___

  3. ___

  4. ___

  5. ___

  1. Reflection over the -axis.

  2. Vertical shift of , making the horizontal asymptote .

  3. Horizontal shift left , making the vertical asymptote .

  4. Vertical stretch by a factor of .

  5. Horizontal shift right , making the vertical asymptote .

11.

Graph each of the following equations without using technology.

a.

a blank 17 by 17 grid

b.

a blank 17 by 17 grid

12.

Describe the features of the function

Vertical asymptote:

Horizontal asymptote:

Vertical stretch factor:

Domain:

Range:

Ready for More?

You have already named the asymptotes and other features of . Now, try naming the anchor points of the function.

Takeaways

Transformations of

Try one more:

a blank 17 by 17 grid

Lesson Summary

In this lesson, we learned to graph functions that are transformations of . 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: .

Retrieval

1.

Find the domain of .

2.

Find all of the roots of , given that is a root of . Then write in factored form.

Roots:

Factored form: