Lesson 2 Piece-Wiser Solidify Understanding

Learning Focus

Graph piecewise functions.

Interpret piecewise functions.

Are all piecewise functions continuous?

Technology guidance for today’s lesson:

Construct a Piecewise Function for a Graph: Casio ClassPad Casio fx-9750GIII

Open Up the Math: Launch, Explore, Discuss

Rashid is off on another bike ride. He has a route he likes to do on his own and has modeled his ride with the following piecewise function to represent the average number of miles he travels in minutes:

$g (x) = {\begin{cases} \frac{1}{4} x, & 0 \leq x \leq 20 \\ \frac{1}{5} (x - 20) + 5, & 20 < x \leq 50 \\ \frac{2}{7} (x - 50) + 11, & 50 < x \leq 92 \\ \frac{1}{8} (x - a) + b, & 92 < x \leq 100 \end{cases}$

1.

What is the domain for this function? What does the domain represent in this context?

2.

What is the average rate of change during the interval $[20, 50]$ ?

3.

The average rate of change is greatest in which time interval?

4.

Find the value of each and explain what the value means in this context.

a.

$g (30) =$

b.

$g (64) =$

c.

$g (10) =$

5.

Complete the last equation by finding values for $a$ and $b$ .

6.

Sketch a graph of $g (x)$ . Label the axes with appropriate units for this context.

7.

Compare the equations $f (x) = \frac{1}{5} (x - 20) + 5$ and $g (x) = \frac{1}{5} {(x - 20)}^{2} + 5$ . What are the similarities and differences?

8.

How does point-slope form for linear functions compare to vertex form for quadratic functions?

Use the piecewise function $h (x)$ to answer the following problems.

$h (x) = {\begin{cases} - (x + 3) + 4, & - 3 \leq x < 0 \\ 1, & 0 \leq x < 3 \\ - 3, & 3 \leq x < 6 \end{cases}$

9.

Sketch the graph of $h (x)$ .

10.

What is the domain of $h (x)$ ?

11.

Find:

a.

$h (- 1)$

b.

$h (5)$

c.

$h (3)$

Ready for More?

Graph the function:

$f (x) = {\begin{cases} 4 - x^{2}, & - 2 \leq x < 1 \\ (x - 1) + 3, & 1 \leq x < 3 \\ 1, & x \geq 3 \end{cases}$

Takeaways

When finding output values for given input values in a piecewise function, you must:

Piecewise functions and point-slope form:

Lesson Summary

In this lesson, we graphed piecewise functions and learned that some are discontinuous. We learned how to indicate on a graph whether the point was included in an interval. We also made connections between point-slope form for a line and vertex form for a quadratic function.

Retrieval

Find the solutions for each equation. (There are two solutions.)

1.

$| x | = 23$

2.

$| w + 8 | = 15$

3.

$2 | x - 3 | - 19 = - 15$

Given the quadratic parent function, $f (x) = x^{2}$ , create a new function in vertex form, $g (x)$ , that fits the description.

4.

Shift $f (x)$ right $7$ and down $19$ .

5.

Shift $f (x)$ left $12$ , reflect vertically, and shift up $28$ .