Lesson 1: Graphs of Piecewise Linear Functions

NYS COMMON CORE MATHEMATICS CURRICULUM

Lesson 1: Graphs of Piecewise Linear Functions

Classwork Example 1

Lesson 1 M1

ALGEBRA I

Lesson 1: Date:

Graphs of Piecewise Linear Functions 8/7/13

? 2013 Common Core, Inc. Some rights reserved.

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

S.1

NYS COMMON CORE MATHEMATICS CURRICULUM

Lesson 1 M1

ALGEBRA I

Example 2

Here is an elevation-versus-time graph of a person's motion. Can we describe what the person might have been doing?

PIECEWISE-DEFINED LINEAR FUNCTION: Given non-overlapping intervals on the real number line, a (real) piecewise linear function is a function from the union of the intervals on the real number line that is defined by (possibly different) linear functions on each interval.

or

Lesson 1: Date:

Graphs of Piecewise Linear Functions 8/7/13

? 2013 Common Core, Inc. Some rights reserved.

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

S.2

NYS COMMON CORE MATHEMATICS CURRICULUM

Lesson 1 M1

ALGEBRA I

Problem Set

1. Watch the video, "Elevation vs. Time #3" (below)

. (This is the third video under "Download Options" at the site called "Elevation vs. Time #3.")

It shows a man climbing down a ladder that is 10 feet high. At time 0 seconds, his shoes are at 10 feet above the floor, and at time 6 seconds, his shoes are at 3 feet. From time 6 seconds to the 8.5 second mark, he drinks some water on the step 3 feet off the ground. Afterward drinking the water, he takes 1.5 seconds to descend to the ground and then he walks into the kitchen. The video ends at the 15 second mark.

a. Draw your own graph for this graphing story. Use straight line segments in your graph to model the elevation of the man over different time intervals. Label your -axis and -axis appropriately and give a title for your graph.

b. Your picture is an example of a graph of a piecewise linear function. Each linear function is defined over an interval of time, represented on the horizontal axis. List those time intervals.

c. In your graph in part (a), what does a horizontal line segment represent in the graphing story?

d. If you measured from the top of the man's head instead (he is 6.2 feet tall), how would your graph change?

e. Suppose the ladder is descending into the basement of the apartment. The top of the ladder is at ground level (0 feet) and the base at the ladder is 10 feet below ground level. How would your graph change in observing the man following the same motion descending the ladder?

f. What is his average rate of descent between time 0 seconds and time 6 seconds? What was his average rate of descent between time 8.5 seconds and time 10 seconds? Over which interval does he descend faster? Describe how your graph in part a can also be used to find the interval during which he is descending fastest.

Lesson 1: Date:

Graphs of Piecewise Linear Functions 8/7/13

? 2013 Common Core, Inc. Some rights reserved.

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

S.3

NYS COMMON CORE MATHEMATICS CURRICULUM 2. Make up an elevation-versus-time graphing story for the following graph:

Lesson 1 M1

ALGEBRA I

3. Draw up an elevation-versus-time graphing story of your own and then make up a story for it.

Lesson 1: Date:

Graphs of Piecewise Linear Functions 8/7/13

? 2013 Common Core, Inc. Some rights reserved.

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

S.4

NYS COMMON CORE MATHEMATICS CURRICULUM

Lesson 2: Graphs of Quadratic Functions

Classwork Example 2

Plot a graphical representation of change in elevation over time for the following "graphing story." It is a video of a man jumping from 36 feet above ground into 1 foot of water. or (If neither link works, search for "OFFICIAL Professor Splash World Record Video!")

Lesson 2 M1

ALGEBRA I

Lesson 2: Date:

Graphs of Quadratic Functions 8/7/13

? 2013 Common Core, Inc. Some rights reserved.

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

S.5

NYS COMMON CORE MATHEMATICS CURRICULUM

Lesson 2 M1

ALGEBRA I

Example 3

The table below gives the area of a square with sides of whole number lengths. Have students plot the points in the table on a graph and draw the curve that goes through the points.

Side (cm)

0

1

2

3

4

Area (cm2)

0

1

4

9

16

On the same graph, reflect the curve across the -axis. This graph is an example of a "graph of a quadratic function."

Lesson 2: Date:

Graphs of Quadratic Functions 8/7/13

? 2013 Common Core, Inc. Some rights reserved.

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

S.6

NYS COMMON CORE MATHEMATICS CURRICULUM

Lesson 2 M1

ALGEBRA I

Problem Set

1. Here is an elevation versus time graph of a ball rolling down a ramp. The first section of the graph is slightly curved.

a. From the time of about 1.7 seconds onwards, the graph is a flat horizontal line. If Ken puts his foot on the ball at time 2 seconds to stop the ball from rolling, how will this graph of elevation versus time change?

b. Estimate the number of inches of change in elevation of the ball from 0 seconds to 0.5 seconds. Also estimate the change in elevation of the ball between 1.0 seconds and 1.5 seconds.

c. At what point is the speed of the ball the fastest, near the top of the ramp at the beginning of its journey or near the bottom of the ramp? How does your answer to part (b) support what you say?

2. Watch the following graphing story: Elevation vs. Time #4 [. This is the second video under "Download Options" at the site called "Elevation vs. Time #4."] The video is of a man hopping up and down several times at three different heights (first, five medium-sized jumps immediately followed by three large jumps, a slight pause, and then 11 very quick small jumps).

a. What object in the video can be used to estimate the height of the man's jump? What is your estimate of the object's height?

b. Draw your own graph for this graphing story. Use parts of graphs of quadratic functions to model each of the man's hops. Label your -axis and -axis appropriately and give a title for your graph.

Lesson 2: Date:

Graphs of Quadratic Functions 8/7/13

? 2013 Common Core, Inc. Some rights reserved.

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

S.7

NYS COMMON CORE MATHEMATICS CURRICULUM 3. Use the table below to answer the following questions.

a. Plot the points (, ) in this table on a graph (except when is 5).

Lesson 2 M1

ALGEBRA I

b. The -values in the table follow a regular pattern that can be discovered by computing the differences of consecutive -values. Find the pattern and use it to find the -value when is 5.

c. Plot the point you found in part (b). Draw a curve through the points in your graph. Does the graph go through the point you plotted?

d. How is this graph similar to the graphs you drew in Examples 1, 2, and 3? Different?

Lesson 2: Date:

Graphs of Quadratic Functions 8/7/13

? 2013 Common Core, Inc. Some rights reserved.

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

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