Lesson 4 Absolutely Transformed Practice Understanding

Learning Focus

Use transformations to graph absolute value functions.

Write the equation that corresponds to the graph of an absolute value function.

How do the graphs of quadratic functions compare to absolute value functions?

Technology guidance for today’s lesson:

Investigate Graph Transformations With Sliders: Casio ClassPad

Open Up the Math: Launch, Explore, Discuss

In Absolutely Valuable, you explored the linear absolute value function: $f (x) = | x |$ . In this task, we’re going to use what we know about transforming functions to become fluent with graphing linear absolute value functions and writing them in piecewise notation.

Let’s get started with identifying some helpful anchor points on $f (x) = | x |$ .

The graph of $f (x)$ is shown. Identify the vertex, line of symmetry, and anchor points.

1.

Vertex:

Line of symmetry:

Anchor points:

$(1, \underset{―}{})$ , $(2, \underset{―}{})$ , $(3, \underset{―}{})$
$(- 1, \underset{―}{})$ , $(- 2, \underset{―}{})$ , $(- 3, \underset{―}{})$

2.

If you had to make an educated guess, what transformation has been applied to $f (x) = | x | + 2$ ?

Draw the graph with three accurate points on both sides of the line of symmetry, check it using technology, and write the function in piecewise form.

Piecewise form:

Use your understanding of transformations and absolute value functions to write the equation of the graphs in piecewise and absolute value form.

3.

Absolute value form:

Piecewise form:

How do these two equations relate to each other?

4.

Absolute value form:

Piecewise form:

How do these two equations relate to each other?

5.

Absolute value form:

Piecewise form:

How do these two equations relate to each other?

You got this! Use transformations to graph each of the following functions and write the equation in piecewise form.

6.

$f (x) = - | x | + 4$

Piecewise form:

7.

$f (x) = \frac{1}{2} | x - 3 | - 1$

Piecewise form:

8.

$f (x) = - 2 | x - 1 | + 5$

Piecewise form:

Graph the following functions and write the equations in absolute value form.

9.

$f (x) = {\begin{aligned} - (x - 4), & x < 4 \\ (x - 4), & x \geq 4 \end{aligned}$

Absolute value form:

10.

$f (x) = {\begin{aligned} - (x) + 1, & x < 0 \\ (x) + 1, & x \geq 0 \end{aligned}$

Absolute value form:

11.

$f (x) = {\begin{aligned} - 3 (x + 2) + 1, & x < - 2 \\ 3 (x + 2) + 1, & x \geq - 2 \end{aligned}$

Absolute value form:

Ready for More?

Not all absolute value functions are linear! Graph $g (x)$ and write it as a piecewise function.

$g (x) = | x^{2} - 4 | + 1$

How is the graph of this function related to the graph of $f (x) = x^{2} - 4$ ?

Piecewise form:

Takeaways

Identifying transformations in absolute value form:

To change from absolute value form to piecewise form:

Lesson Summary

In this lesson, we learned the quick-graph method for graphing absolute value functions using transformations. We learned to change from absolute value to piecewise form and to identify the vertex and line of symmetry from either form.

Retrieval

Solve each equation for the indicated variable.

1.

Solve for $y$ .

$- 8 x + 2 y = 20$

2.

Solve for $b$ .

$A = \frac{b h}{2}$

Simplify each expression; write in simplest radical form.

3.

$(7 + 2 \sqrt{5}) - (12 - 4 \sqrt{5})$

4.

$\sqrt{8} + \sqrt{18} - \sqrt{50}$