Name: ______Date: ______Hr: ______
Ch 11 WS #2 Exponential Functions vs. Power Function
Power Functions and Exponential Functions look similar, but they are actually quite different.
Compare Graphs:
1. Compare the graphs of y = x2 and y = 2x
Fill in each chart. Then graph each on the axes provided. Label your axes with appropriate numbers. List the domain and range of each graph below.
(a) y = x2 (b) y = 2x
x / -2 / -1 / 0 / 1 / 2 / x / -2 / -1 / 0 / 1 / 2y / y
D: ______R: ______D: ______R: ______
2. Compare the graphs of y = x3 and y = 3x
Fill in each chart. Then graph each on the axes provided. Label your axes with appropriate numbers:
(a) y = x3 (b) y = 3x
x / -2 / -1 / 0 / 1 / 2 / x / -2 / -1 / 0 / 1 / 2y / y
D: ______R: ______D: ______R: ______
3. Given a graph of y = xn and y = nx, how could you tell which graph is which? Explain.
4. Compare the graphs of y = x1/2 and y = ( ½ )x
Fill in each chart. Then graph each on the axes provided. Label your axes with appropriate numbers:
(a) y = x1/2 (b) y = ( ½ )x
x / -2 / -1 / 0 / 1 / 2 / x / -2 / -1 / 0 / 1 / 2y / y
D: ______R: ______D: ______R: ______
(c) How are the graphs the same? How are they different?
5. Compare the graphs of y = 5∙x2 and y = 5∙2x
Fill in each chart. Then graph each on the axes provided. Label your axes with appropriate numbers:
(a) y = 5∙x2 (b) y = 5∙2x
x / -2 / -1 / 0 / 1 / 2 / x / -2 / -1 / 0 / 1 / 2y / y
D: ______R: ______D: ______R: ______
(c) How are the graphs the same? How are they different?
Power Function: y = k∙xn where k & n are constants
Examples: y = x2 y = -3x5 y = 3 x 1/2
Exponential Function: y = k∙nx where k & n are constants
Examples: y = 2x y = -3∙(5)x y = 3 ( ½ )x
6. Look at the examples above. What is the difference between a power function and an exponential function? Explain.
7. Look at all of the functions listed in questions 1, 2, 4, & 5. List all those that are…
(a) Power Functions:
(b) Exponential Functions:
Graphs of Exponential Functions: Growth vs. Decay:
Exponential functions are all of the form y = k∙nx. Often they are thought of as functions of time and thus written y = k∙nt. Complete the following graphs to try and figure out what the ‘k’ value and the ‘n’ value tell you.
8. y = 5∙(1.2)t 9. y = 5∙(0.8)t
t / 0 / 1 / 2 / 3 / 4 / t / 0 / 1 / 2 / 3 / 4y / y
10. y = 10∙(1.25)t 11. y = 10∙(0.75)t
y / y
12. y = 2∙(1.5)t 13. y = 2∙(0.5)t
t / 0 / 1 / 2 / 3 / 4 / t / 0 / 1 / 2 / 3 / 4y / y
14. y = 4∙(3)t 15. y = 4∙(0.3)t
t / 0 / 1 / 2 / 3 / 4 / t / 0 / 1 / 2 / 3 / 4y / y
16. Sometimes exponential functions grow as time goes on, and sometimes they decay. Which values in the formula y = k∙nt tells you whether the function grows or decays: the ‘k’ value or the ‘n’ value? Explain.
17. Challenge Question: How can you use the formula y = k∙nt to tell you the actual percentage of growth each day or year? How can you use the formula y = k∙nt to tell you the actual percentage of decay each day or year? Explain.
18. In the formula y = k∙nt , what does the ‘k’ value tell you? Explain.
19. The city leaders in Camelot are having a disagreement about population growth. They have graphed the population vs. time since the year 1990 and got a graph that curves up as pictured at right.
The mayor believes that the growth can be modeled by a normal parabola while the city manager insists it is an exponential graph.
(a) Explore what will happen if the mayor is correct and population follows the formula:
P = 3t2 + 30 where population is in thousands
and time is in years, year zero is 2005
Fill in the table if the mayor’s formula is the true formula:
t (yrs) / 0 / 1 / 2 / 6 / 4 / 5 / 6 / 7 / 8P (thousands)
t (yrs) / 9 / 10 / 11 / 12 / 13 / 14 / 15 / 16 / 17
P (thousands)
(b) Explore what will happen if the city manager is correct and population follows the formula:
P = 30(1.30)t where population is in thousands
and time is in years, year zero is 2005
Fill in the table if the city manger’s formula is the true formula:
t (yrs) / 0 / 1 / 2 / 6 / 4 / 5 / 6 / 7 / 8P (thousands)
t (yrs) / 9 / 10 / 11 / 12 / 13 / 14 / 15 / 16 / 17
P (thousands)
(c) In terms of planning Camelot city facilities over the next 20 years, what will be the difference between the mayor’s conjecture and the city manager’s conjecture about population growth? Which growth model would be better supported by city services?
(d) You are considering moving to Camelot. Would you be more interested if the population was going to follow the mayor’s formula and be a parabola? Or would you prefer it to follow the exponential model? Explain.
20. Human population actually has followed an exponential growth model as opposed to the parabolic model. Why might this be a concern in terms of resources such as energy and food? Explain.