Algebra
Chapter 9, Answers
Quadratic Functions and Equations
What do the Leaning Tower of Pisa and mathematics have in common? Let's investigate!
Part A, Graphing Quadratic Functions
1. Galileo was a very interesting man. For a brief overview of his life,
visit the Galileo
Web site. Switch to Galileo's
Experiments. Then run the experiment at that site by clicking on
the inclined plane. Gather data for the total distance covered at one second
intervals.
[Data will vary.]
2. What is the relationship between the total distance covered and time? [Total distance covered is proportional to time squared.]
3.Graph your data.
[We're sorry for the inconvience, but the graph
is not yet available. Please check this site again.]
a.What shape is the graph?
[The graph is the right side of a parabola.]b.Why does it not make sense to see data in Quadrant II?
[It is not possible to have negative values for time.]c.Why does the site say distance is proportional to time squared?
[There can be a constant coefficient in front of time squared.]
Part B, Modeling with Quadratic Functions
4. Continue to use the Galileo site. Rewrite the distance equation in
the form h = 1/2 at2 + vt + s.
[h = 1/2 ( -32 ft/sec)2 +(0) t + 0.]
5. In the above equation, why is a (acceleration) equal to -32?
[This represents the effect of the pull of gravity
on an object.]
6. Make a table for 1 second, 2 seconds, 3 seconds, and 4 seconds. Use
the equation to calculate the height.
Time in seconds |
Height in feet |
1 |
16 |
2 |
64 |
3 |
144 |
4 |
256 |
a. Graph you equation using a graphing calculator.
[We're sorry for the inconvience, but the graph is not yet available. Please check this site again.]b. Describe your graph.
[The graph is a parabola with vertex (0,0) and opening downward.]
Part C, Making Connections
7. Work with a partner. Make a model of Galileo's inclined plane experiment using a cardboard ramp as described in your text.
a. Gather the data for a 15-degree ramp and a 30-degree ramp.
[Answers will vary.]b. Make a table of your data from 0, 1, 2, 3, and 4 even intervals on your ramp.
[Answers will vary.]c. Write the equation that fits your data.
[Answers will vary.]d. Graph this function.
[Graph is a parabola.]
8. Compare your work with other groups. What is similar? What is different?
[Answers will vary.]
How can the graph of a quadratic equation help us to answer questions about the height of an object? Let's continue to investigate!
Part A, Quadratic Functions and Equations
1. Look again at the Galileo's Experiements Web site.
a. The expression d ~ t2 means d is proportional to t2. Suppose the ratio d divided by t2 is equal to 5. Write the equation for d in terms of t.
[d = 5t2]b. Graph the equation.
[We're sorry for the inconvience, but the graph is not yet available. Please check this site again.]c. Where does the graph cross the x-axis?
[The graph crosses the x-axis at the origin or (0,0).]
2. Suppose the height of a golf ball is described after initial velocity from the ground of 4.9 m/sec.
a. Write the equation for describing the height of the ball after t seconds have elapsed.
[ h = 1/2 (- 9.8 m/sec) (t2) + 4.9 t + 0 ]b. By graphing, predict how long it will be before the ball returns to the ground?
[The height equals 0 after 1 second.]
Part B, Solving Using Tables and Factoring
3. Solve your equation from 2a by factoring.
[0 = 1/2 ( 32 ft/sec)2 +(4.9) t + 0. Continuing,
0 = -4.9t (t - 1). Therefore, the ball will be on the ground
after 1 second.]
a. Why does it make sense to have time equal to 0?
[Time is zero before the ball leaves the ground.]b. When does the golf ball reach its highest point?
[It reaches its highest point after 0.5 seconds.]c. How high will the golf ball be at that time?
[The ball will be 1.225 meters above the ground after .5 seconds.]d. Would you want to play golf on the same team with this person? Explain.
[I would only want to play with this person if he/she only had to shoot 4.9 meters to reach the cup.]
Part C, Making Connections
4. Name three methods you can use to solve quadratic equations.
[Graphing, using tables, factoring]
5. Describe the limitations of each of these methods.
[Sample answers--
Graphing: Some values are very difficult to read on a graph.
Using tables: Not all values are listed on a table. The value you want may
not be there.
Factoring: Not all quadratic equations are factorable.]
Sometimes graphing, using tables, and factoring cannot lead us to exact
solutions to a quadratic equation. How else can we get the solution? Let's
investigate!
Part A, Solving with Square Roots
1. Look at the Stockmaster Web site.
a. Choose any three mutual funds. For each fund, look at the earnings chart and record the approximate cost of the fund on January 1, 1998.
[Answers will vary.]b. What would the interest rate be if you invested $1000 and earned $1200 assuming the interest compounded annually for a two year period?
[Using the formula A = P ( 1 + r) 2, the interest rate would be 9.54%.]
Part B, The Quadratic Formula
2. Look back to the three mutual funds you investigated above. Calculate
the percent return on your three funds for this year.
[Answers will vary.]
a. How much interest would be earned in a 2-year period if $1000 were invested in each of your funds and they compounded annually?
[Answers will vary.]b. If $300 were earned over a 2-year period and your funds compounded annually, use the quadratic formula to calculate the rate of return on your $1000 investment.
[Answers will vary.]
Part C, Making Connections
3. Explain why the quadratic formula can be more useful than other methods
for solving quadratic equations.
[The quadratic formula can be applied to any quadratic
equation; it works all the time.]