Unit 5, Lesson 10
Piecewise Linear Functions
Grade 8
10.1
Warm-up
What do you notice? What do you wonder?
Scatterplot, horizontal, time in hours after midnight, 0 to 12 by ones, vertical, temperature in degrees Fahrenheit. Fifty points approximate a straight line from point 2 5 comma 50 increasing to 5 point 75 comma 59 and then decreasing from there to 12 comma 52 point 5.
10.2
Activity
A scatterplot, horizontal, year, 1900 to 2015 by fives, vertival, percentage recycled, 15 % to 27 % by threes. Beginning with 1991 comma 16%, points trend linearly upward to 1996 comma 22%, then linearly upward at a less steep rate to 2011 comma 26 %, then two points trend down.
- Approximate the percentage recycled each year with a piecewise linear function by drawing between 3 and 5 line segments to approximate the graph.
- Find the slope for each piece. What do these slopes tell you?
10.3
Activity
Elena filled up the tub and gave her dog a bath. Then she let the water out of the tub.
- The graph shows the amount of water in the tub, in gallons, as a function of time, in minutes. Add labels to the graph to show this.
- When did she turn off the water faucet?
- How much water was in the tub when she bathed her dog?
- How long did it take for the tub to drain completely?
- At what rate did the faucet fill the tub?
- At what rate did the water drain from the tub?
10.4
Activity
The graph shows the speed of a car as a function of time. Describe what a person watching the car would see.
Student Lesson Summary
This graph shows Andre biking to his friend’s house, where he hangs out for a while. Then they bike together to the store to buy some groceries before racing back to Andre’s house for a movie night. Each line segment in the graph represents a different part of Andre’s travels.
Graph composed of 5 linear sections. Horizontal axis, time, vertical axis, distance from home. Beginning at the origin, first segment slopes up as it moves right. Second segment horizontal, third segment slopes up and right, more steeply than the first segment. Fourth segment horizontal, fifth segment slopes down and right back to the horizontal axis.
This is an example of a piecewise linear function, which is a function whose graph is pieced together out of line segments. It can be used to model situations in which a quantity changes at a constant rate for a while, then switches to a different constant rate.
We can use piecewise functions to represent stories, or we can use them to model actual data. In the second example, temperatures recorded at different times throughout a day are modeled with a piecewise function made up of two line segments. Which line segment do you think does the best job of modeling the data?
Scatterplot, horizontal, time in hours after midnight, 0 to 12 by ones, vertical, temperature in degrees Fahrenheit. Fifty points approximate a straight line from point 2 5 comma 50 increasing to 5 point 75 comma 59 and then decreasing from there to 12 comma 52 point 5.
None