Algebra
Chapter 5 Answers
Analyzing Linear Functions
and Their Graphs
The Great Pyramids have fascinated people for generations. You can use the Great Pyramids to investigate the mathematical concept of slope. Throughout this activity, you will be writing slope using decimals rather than the fractions you are used to seeing. Just keep in mind that nothing changes: Slope is still rise over run or vertical distance over horizontal distance.
Part A, Exploring Slope
1. Try building a scale model at the Great
Pyramid Web site. If the scaled height is 4.9 cm, estimate the slope
of the side of the pyramid. Explain how you figured out the answer.
[Since slope is the change in y divided
by the change in x, the slope of the pyramid is approximately 4.9/3.85
or 1.27. Here, 4.9 is the height of the pyramid, and 3.85 is half the length
of one of the sides.]
Part B, Rate of Change
2. Refer to the pyramid model from 1. Imagine that you are looking
at the pyramid from the side. Assign coordinates to the vertex of the pyramid
and to the bottom left side of the base of one face of the pyramid. Write
your coordinates in scaled form or centimeters.
[Possible solutions include: vertex (0, 4.9) and
base (-3.85, 0).]
Part C, The Geometry of Slope
3. Using your pyramid model, make a conjecture about the four faces of
the pyramid. Explain your conjecture.
[Answers will vary. One possible answer is that
the faces are the same shape and the same size since the base of the pyramid
is a square and the height of each face is the same.]
Part D, Making Connections
7. Using information from the Scale
Model of the Great Pyramid Web page, create a scale model to represent
Khafre and Menkaure.
[Models will vary.]
8. Find the slope of the sides of Khafre and Menkaure.
[Khafre has a slope of 1.34; Menkaure has a slope
of 1.20]
9. Of the three pyramids, which one is the steepest? Which one is the
flattest? Explain how you know.
[Khafre is the steepest because it has the greatest
rate of change or slope (1.34) and Menkaure is the flattest with a rate
of change of 1.20.]
Health and fitness are important ideas for all of us to consider. Using the information from this superlesson and data from the World Wide Web, you can calculate personal fitness statistics.
Part A, Slope and Location of Lines
1. Browse through Focus on Your Family's Health Web site for information about cardiovascular exercise.
a. Calculate your maximum heart rate according to the steps provided.
[Answers will vary. The table shows averages.]
Average 13-year old
176 heart beats per minute
Average 14-year old
175 heart beats per minute
Average 15-year old
174 heart beats per minute
Average 16-year old
173 heart beats per minute
b. Find your resting heart rate. [Answers will vary.]
c. If you were to exercise for 5 minutes to achieve your maximum heart rate, graph a line showing your starting heart rate and your heart rate after 5 minutes. Assume that your heart rate would increase linearly over the 5-minute period. [Graphs will vary.]
d. Find the slope of the line containing the points that represent your resting heart rate and your maximum heart rate after exercising for 5 minutes.
[Answers will vary but the slope will be positive.]e. Where does your graph cross the y-axis?
[Graphs will vary but the x-coordinate will be 0 and the y-coordinate will be the student's resting heart rate.]
Part B, Two-Point Equation of a Line
2. Target heart rates show exercisers what they should try to maintain as they exercise.
a. Calculate your target zone for beginner level and intermediate level.
[The target zone for beginner level is 60-65% of the student's maximum heart rate. The target zone for intermediate level is 70-75% of the student's maximum heart rate. A 14-year old has a maximum heart rate of about 175. So the beginner level is 105-114; the intermediate level is 122-131.]b. If you were to exercise for 5 minutes to achieve your target heart rate, write an equation of that line in slope-intercept form. That line should contain your resting heart rate as well as your heart rate target zone minimum value. [Answers will vary.]
Part C, Slope and Dimensional Analysis
3. Use the information from 2 to complete the following.
a. Re-graph your line three times using different scales on the vertical axis but keeping the size of your graph paper the same. [Graphs will vary.]
b. What can you conclude about the slope of the three lines?
[Even though the lines are the same, the scale affects the appearence of the slope. To compare the graphs visually, corresponding scales on each graph must be the same.]
Part D, Scatter Plots and Trend Lines
4. Survey twenty people of different ages and have them calculate their resting heart rate.
a. Graph the survey information using age on the horizontal axis and heart rate on the vertical axis. (A graphing calculator may be used.) [Graphs will vary.]
b. Draw in the trend line or line of best fit.
[Graphs will vary.]c. Write the equation of your line of best fit.
[Answers will vary.]
Part E, Making Connections
5. Graphing allows you to combine your data.
a. Calculate your resting heart rate and your target heart rate zone. [Answers will vary.]
b. After doing some mild stretches, walk briskly for at least 20 minutes. Take your pulse every 3-5 minutes during exercise to make sure you are in your target zone. Record this information. [Answers will vary.]
c. Record your heart rate during your cool down activities and then again 10 minutes after cool-down.
[Answers will vary.]d. Graph your heart rate information using time on the horizontal axis and heart rate on the vertical axis.
[Graphs will vary.]e. Write an equation of the line indicating that your heart rate is decreasing during cool down. [Answers will vary.]
f. If you were to exercise 3-5 times per week, how do you think your graph would change? Predict how your graph from part e might change.
[Predictions will vary, but one prediction may be that the time to achieve the resting heart rate after exercise may decrease.]