Lesson 10
Piecewise Linear Functions
Let’s explore functions built out of linear pieces.
10.1: Notice and Wonder: Lines on Dots
What do you notice? What do you wonder?
![Scatterplot, horizontal, time in hours after midnight, 0 to 12 by ones, vertical, temperature in degrees Farenheit.](https://cms-im.s3.amazonaws.com/LNkEzPX7tPB7bkTUCNdhSFxS?response-content-disposition=inline%3B%20filename%3D%228-8.5.C3.Image.03.png%22%3B%20filename%2A%3DUTF-8%27%278-8.5.C3.Image.03.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T004135Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=9c7a484a368ff7a229ee8acb07d07851d287a0e1fdf6c49197605bf64075d0ef)
10.2: Modeling Recycling
![A scatterplot, horizontal, year, 1900 to 2015 by fives, vertival, percentage recycled, 15 % to 27 % by threes.](https://cms-im.s3.amazonaws.com/9qRk7zjsCPdMRbYTpUDyEVWv?response-content-disposition=inline%3B%20filename%3D%228-8.5.B3.Image.03.png%22%3B%20filename%2A%3DUTF-8%27%278-8.5.B3.Image.03.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T004135Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=d6c0c5a739869c6fb12e57a8fa1ef83190091c580caa5ffddf86dbb5d4e35d86)
- Approximate the percentage recycled each year with a piecewise linear function by drawing between three and five line segments to approximate the graph.
- Find the slope for each piece. What do these slopes tell you?
10.3: Dog Bath
Elena filled up the tub and gave her dog a bath. Then she let the water out of the tub.
![Coordinate plane, horizontal, 0 to 30 by threes, vertical, 0 to 30 by threes. Graph consists of 3 connected segments. First, origin to (6 comma 24). Next to (18 comma 24), then to (30 comma 0).](https://cms-im.s3.amazonaws.com/HqQEkzP2tDMpmKqF9hHchLej?response-content-disposition=inline%3B%20filename%3D%228-8.5.C.10.30.png%22%3B%20filename%2A%3DUTF-8%27%278-8.5.C.10.30.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T004135Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=b284547d60da5b352b475f75c18b6654bdd0e5732f72cc55f3ac7448534ef196)
- 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: Distance and Speed
![Graph, horizontal, time, vertical, speed. The graah is composed of three line segments. First, from the origin and sloping up, second, horizontal, and third sloping down to the horizontal axis.](https://cms-im.s3.amazonaws.com/HLPDmL1QbWQm6kMSqCVEjUvt?response-content-disposition=inline%3B%20filename%3D%228-8.5.C3.Image.05.png%22%3B%20filename%2A%3DUTF-8%27%278-8.5.C3.Image.05.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T004135Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=5f04ebe64d15e1dd8d7a5f7a3b54d77964f47f9a9c1432a1579bdf89af24bc96)
The graph shows the speed of a car as a function of time. Describe what a person watching the car would see.
The graph models the speed of a car over a function of time during a
3-hour trip. How far did the car go over the course of the trip?
There is a nice way to visualize this quantity in terms of the graph. Can you find it?
![Coordinate plane. time in minutes by speed in miles per hour.](https://cms-im.s3.amazonaws.com/388yEGVsmKKXR19G8zvRCoBT?response-content-disposition=inline%3B%20filename%3D%228-8.5.10.Area.png%22%3B%20filename%2A%3DUTF-8%27%278-8.5.10.Area.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T004135Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=f34ca58dd1c810be626161910c07c38dd19a9531c376d1695a7bc4dcd8a92531)
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.](https://cms-im.s3.amazonaws.com/JrMjYfyHo5eqgjbAPsmLG8k1?response-content-disposition=inline%3B%20filename%3D%228-8.5.C10.Image.Revision.112.png%22%3B%20filename%2A%3DUTF-8%27%278-8.5.C10.Image.Revision.112.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T004135Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=8f590e97b123aa07b46de5e9ba7be7b5d6bdfd3634316f30e06238a20fae27b5)
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, temperature recordings at several 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 Farenheit.](https://cms-im.s3.amazonaws.com/LNkEzPX7tPB7bkTUCNdhSFxS?response-content-disposition=inline%3B%20filename%3D%228-8.5.C3.Image.03.png%22%3B%20filename%2A%3DUTF-8%27%278-8.5.C3.Image.03.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T004135Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=9c7a484a368ff7a229ee8acb07d07851d287a0e1fdf6c49197605bf64075d0ef)