Lesson 1 Going to Pieces Develop Understanding

Learning Focus

Relate the graph of a function to a story context.

Write a function made up of several functions.

How can I write one function that is made up of pieces of different functions?

Open Up the Math: Launch, Explore, Discuss

Rashid has a part-time job as a pizza delivery driver. As part of the job, he must keep track of the time and distance travelled to each stop. Sometimes he delivers to more than one customer in a trip. The graph given shows one of his trips.

1.

Tell the story of Rashid’s trip. Include the distance and the time for each leg of the journey in your story.

2.

Write the equations of the lines that model each piece of the trip.

Segment $1$ :

Segment $2$ :

Segment $3$ :

Segment $4$ :

Pause and Record: Adding Notation, Vocabulary, and Conventions

The function you created above is called a piecewise function. In mathematics, a piecewise-defined function is a function defined by more than one sub-function (or piece of a function), with each section only existing in a certain interval of the function’s domain.

Michelle and Rashid love going on long bike rides. Every Saturday, they have a particular route they bike together that takes four hours. Given is a piecewise function that estimates the distance they travel in kilometers for each hour of their bike ride.

$f (x) = {\begin{cases} 16 x, & 0 \leq x \leq 1 \\ 10 (x - 1) + 16, & 1 < x \leq 2 \\ 14 (x - 2) + 26, & 2 < x \leq 3 \\ 12 (x - 3) + 40, & 3 < x \leq 4 \end{cases}$

3.

What part of the bike ride are they going the fastest? Slowest?

4.

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

5.

Find $f (2)$ . Explain what this means in terms of the context.

6.

How far have they traveled at $3 hours$ ? Write the answer using function notation.

7.

What is the total distance they travel on this bike ride?

8.

Sketch a graph of the bike ride as a function of distance traveled over time.

Ready for More?

Write your own continuous piecewise function with at least three different sub-functions.

How did you ensure that the function was continuous?

Takeaways

Working with piecewise functions:

Adding Notation, Vocabulary, and Conventions

$f (x) = \begin{array}{l} sub-function & domain \\ {\begin{cases} , \\ , \\ , \\ , \end{cases} \end{array}$

Vocabulary

piece-wise defined function
sub-function
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we learned about piecewise functions, functions that combine several pieces that each have their own equation into one function. We graphed and wrote equations for piecewise functions. We learned that the equations for each part of the function are called sub-functions, each with their own domain that tells what part of the piecewise function they define.

Retrieval

1.

a.

$f (- 5)$

b.

$f (0)$

c.

$f (4)$

d.

$f (x) = 3$

Use point-slope form to write the equation of a line based on the given information.

2.

Slope $= 5$ , through point $(3, 12)$

3.

Through the points $(- 7, 10)$ and $(1, - 6)$