Advanced Algebra
Chapter 2 Answers
Linear Functions, Equations, and Inequalities
Cable television has been a booming business in the United States for more than a decade. As the number of cable customers has grown, so have companies' rates and expenditures. Explore some trends in the cable television business using linear functions and equations.
Part A, Direct Variation and Linear Functions
1a. Graph the cable companies' monthly pay rate for the years 1982 to 1997.
b. Is the relationship you see in 1a approximately linear? [No]
2a. Graph the cable companies' programming expenditures for the years 1986 to 1993.
b. Is relationship you see in 2a approximately linear? [Yes]
3a. Pick two points from 2a and assume they define a linear function that describes cable companies' expenditures. Find the slope of the line between those two points. [Answers will vary.]
b. Pick two other points from 2a and find the slope of the line between those two points. [Answers will vary.]
c. Are your answers to 3a and 3b close? Explain what this means. [If students say the answers are close, they should then say that the line is approximately linear. If students say the answers are not close, they should say the line is not approximately linear.]
Part B, Solving Equations
4. Use the graph you made in question 2 to predict what cable companies' expenditures will be in the year 2000. [$6 billion]
5. Suppose the equation to model the cable companies' expenditures could be written as: y = 281x - 556,033. Solve this equation to calculate the expenditures in the year 2000. [$5.967 billion]
6. How close are your answers to questions 4 and 5? [They are fairly close.]
Part C, Analyzing Equations of Lines
7. Cable companies' basic revenue from subscriber services for the years 1986 to 1995 is a fairly linear function.
a. Graph this line.
b. Pick two data points and use them to calculate the slope of the line that describes these data. [The slope is approximately 1330.]
8a. Use the slope you just calculated and the point (1986, 4891) to write the equation of the line in point-slope form. [y - 4891 = 1330 (x - 1986)]
b. Rewrite your answer to 8a in slope-intercept form. [y = 1330 x - 2636489]
Part D, Making Connections
9. Sunflower Cablevision is a cable provider in Kansas. Read about their set up and monthly rates.
a. How much does basic service cost each month? [$22.95]
b. How much does basic cable installation cost? [$25]
c. Write an equation that relates t, the total cost of service, and n, the number of months of service. [t = 22.95n + 25]
d. Use your equation in 9c to calculate how many months a family can get cable for $300. [About 12 months]
Scatter plots display data visually. A scatter plot helps you to see correlations among data, allowing you to make predictions about other data that conform to the same pattern.
Part A, Using Equations of Trend Lines
1. Find the list of tallest mountains in New Hampshire's White Mountains National Forest (WMNF).
a. Make a scatter plot of the heights of the 9 tallest peaks. Use the coordinates (ranking, height) where ranking is a peak's position on the list from 1 to 9, and height is a peak's height in feet.
b. Draw a trend line for your data.
c. Write an equation for your trend line in the form y = kx + b, where x is height in feet and y is ranking. [Equations will vary but should be approximately y = 150 x + 6200]
d. According to your equation, how high is the 2nd highest peak? the 3rd highest? the 9th highest? [Heights will vary; 5900 ft; 5750 ft; 4850 ft]
e. How well does your equation predict the height of the highest peaks? [Students' equations should predict heights moderately well.]
f. Do you think that the equation will predict the heights of all the peaks in New Hampshire's Hampshire's White Mountain National Forest accurately? Explain. [Answers will vary. Students may argue that the equation will predict the heights of many peaks with some accuracy. They should point out, however, that at some point, the equation will begin to fail as a predictor. That is where the trend line crosses the horizontal axis and begins to predict negative heights. At x = 50, for example, y = height = -150 (50) + 6200 = 1300 ft.]
Part B, The Median-Median Line of Fit
2a. Find the heights of the 9 tallest mountains in New Hampshire's White Mountains National Forest (WMNF). Write the data as coordinate pairs (ranking, height), where ranking is a peak's position on the list from 1 to 9, and height is a peak's height in feet. [ (1, 6288), (2, 5774), (3, 5712), (4, 5382), (5, 5367), (6, 5260), (7, 5089), (8, 4902), (9, 4832) ]
b. Find the three median points A, B, and C. [ (2, 5774), (5, 5367), (8, 4902) ]
b. Find the coordinates of the centroid D of ABC. [ (5, 5347 2/3]
c. Find the equation of the median-median line. Round values to the nearest whole numbers. [ y = 145 x + 6074 ]
d. According to your equation, how high is the 2nd highest peak? the 3rd highest? the 9th highest? [ 5784 ft; 5639 ft; 4769 ft ]
e. How well does your equation predict the height of the highest peaks? [ Moderately well. The predicted height of the 2nd highest peak is within 10 feet of the actual height. The predicted heights of the 3rd and 9th highest peaks are less accurate (within 73 ft and 63 ft, respectively) but still within 2% of the actual figures. ]
f. Find the lengths of the world's 9 longest rivers. Write the data as coordinate pairs (ranking, length), where ranking is a river's position on the list from 1 to 9, and length is a river's length in miles. [ (1, 4145), (2, 4000), (3, 3900), (4, 3740), (5, 3395), (6, 3362), (7, 3030), (8, 2900), (9, 2800) ]
g. Find the equation of the median-median line for the world's 9 longest rivers. [ y = -183 1/3) x + 4348 1/3) ]
Part D, Making Connections
3. Go to the Edmund's Used Car page to look at a selection of used vehicles.
a. Pick your favorite kind of car and make a table of data points (year, price) for the last 9 years. Don't forget to use the same model of car each time. Note: For cars that have fewer than 9 years of data, use only 6 years. [Answers will vary.]
b. Make a scatter plot of your data. [Answers will vary.]
c. Find the equation of the median-median line. [Answers will vary.]
d. Use your data to predict the price of a 10-year-old model of your car. [Answers will vary.]
e. Check the actual data to see how close you were. [Answers will vary.]
Planets have elliptical orbits, not perfectly circular ones. The distance of a planet to the sun is often expressed as an average of the closest and farthest a planet gets from the sun.
Part A, One-Variable Inequalities
1. Gather some information about the planet Jupiter and use it to answer the questions below. Note: You will need to scroll down to "Orbital Parameters."
a. What is the closest Jupiter gets to the sun (perihelion orbit)? [740,600,000 km]
b. Write an inequality expressing this distance. [distance ≥ 740,600,000 km]
c. Graph your inequality from 1b on a number line.
d. What is farthest Jupiter gets from the sun (aphelion orbit)? [816,000,000 km]
e. Write an inequality expressing this distance. [distance ≤ 816,000,000 km]
f. Graph your inequality from 1e on a number line.
g. Write a compound inequality that expresses the distance Jupiter can be from the sun. [740,600,000 ≤ distance ≤ 816,000,000]
h. Graph your inequality from g on a number line.
2. Pick another planet and look at its fact sheet.
a. Write a compound inequality that expresses the distance your planet can be from the sun.
b. Graph your inequality from 2a on a number line.
Part B, Inequalities and Absolute Value
3. Gather some information about the moon and use it to answer the questions below.
a. Suppose the measurement for the moon's surface gravity is only correct to within 5%. What is the maximum possible surface gravity on the moon? [1.701]
b. What is the minimum possible surface gravity of the moon? [1.539]
c. Write an absolute value inequality expressing the surface gravity (g). [| m - 1.62 | = 0.081]
4a. What is the equatorial radius of the moon? [1738 km]
b. What is the closest the moon gets to the Earth during the perigee orbit? [363,300 km]
c. Suppose you don't know if the orbit of the moon is measured from the surface or from the center. What is the closest the surface of the moon could be to the Earth? [361,562 km]
d. What is the farthest the surface of the moon could be from the Earth during the perigee orbit? [363,300 km]
e. Write an absolute value inequality expressing the distance (d) from moon to the Earth at its closest approach. [| d - 362,431 | ≤ 869]
Part C, Linear Inequalities with Two Variables
5. Go to NASA's Planetary Fact Sheet. For each planet, find out the perihelion orbit and the aphelion orbit.
a. Use those numbers to make a table.
b. Graph the data from your table.
c. Draw in a line showing where the orbits would be perfectly circular.
d. Do any planets have perfectly circular orbits? [No, although several are fairly close.]
e. Shade in the area where the perihelion orbit is shorter than the aphelion orbit.
f. Do any of the planet lie in the region you shaded in e? Explain. [Yes, they all do. They must all lie in this region by definition because the Perihelion Orbit is always shorter than the Aphelion Orbit.]
Part D, Making Connections
6. Look at the Mars Fact Sheet to find the distance from Mars to the Sun.
a.Write a compound inequality expressing the distance from Mars to the Sun. [206,600,000 km < distance < 249,200,000 km]
b. Look at the Earth Fact Sheet. Write a compound inequality expressing the distance from Earth to the Sun. [147,100,000 km < distance < 152,100,000 km]
c. What is the farthest Earth is from Mars? [401,300,000 km]
d. What is the closest Earth is to Mars? [59,500,000 km]
7. If it costs NASA $1 billion to send a rocket to Mars when it is at its closest point, and $2.5 billion to send a rocket to Mars when it is at its farthest point, how many of each type of mission could it fund for $15 billion? Express your answer in a graph.