MCR3U Unit 4 Test Review: Exponential Functions
Part A: Knowledge & Understanding
1. Write as a single, positive power and evaluate. Express your answer in rational form.
a) 8-3 x 8
b) (10-2)3
c) 5-7 x 57
d) 45 ÷ 43
e)
(-3)13
(-3)3 x (-3)4
2. Simplify. Use positive exponents in your final answers.
a) (-3x2y2)3(-3x-3y)2
b)
(
)
c)
(
(
)
)
3. What type of relation is represented by the table below? Show any necessary calculations
and explain your reasoning.
x
y
1
2
2
4
3
8
4
16
5
32
6
64
4. Given the f(x) = 0.5(2x-4),
a) State the base function.
b) Describe the transformations that have been applied to the base function in order to
obtain f(x).
c) State the domain, range, y-intercept, and equation of the asymptote for f(x).
Part B: Application
1. Simplify, then evaluate each expression for x = 2 and y = 3. Leave your answers in
rational form.
a)
) /
(
(
) /
b)
√
√
2. There are 400 g of Colbalt-60 in a sample. If Cobalt-60 has a half-life of 2 weeks,
a) Write an equation to represent the amount of Cobalt-60, f(x), after x weeks.
b) Use your equation to determine the amount of Cobalt-60 that has decayed after 8
weeks.
3. The population of a city is growing at an average rate of 3.5% per year. In 1992, the
population was 4200.
a)
b)
c)
d)
Write an equation that models the growth the of the city.
Use your equation to determine the population of the city in 2012.
Determine the year in which the population will have tripled.
In what month of that year will the population have tripled?
Part C: Thinking/Problem Solving
1. Consider the function f(x) = 2-2x+3. Explain and show mathematically why it models
exponential decay.
2. A biologist tracks the population of a new species of frog over several years. The data is
shown in the table below.
Year
Population
0
400
1
480
2
576
3
691
4
829
5
995
a) Graph the data from the table.
b) Does your graph represent exponential growth or decay? How do you know?
c) Estimate the annual growth/decay factor for this relation. Explain or show how you came
up with your estimate.
d) Use your graph to estimate the number of years when the population triples.
Answers
A.
1.
2.
3.
4.
a) 1/64
b) 1/1000000
c) 1
d) 16
e) 729
8
3
5
a) -243y
b) 1/3125a b
c) 8/x
The y values have a common ratio of 2 so it is exponential. (You will need to show this)
a) y = 2x
b) vertical compression by factor of ½, horizontal translation 4 units right
c) d: {x ϵ R}, r: {y ϵ R| y > 0}, y-int = 1/32, asymptote: y = 0
B.
1. a) 27/2
b) 3/2
2. a) f(x) = 400(1/2)x/2
b) 375g
3. a) P(t) = 4200(1.035)t
b) 8357
c) Approx. 31 years
C.
1. Horizontal reflection (across y-axis) -> y values will decrease.
2-2x is equivalent to (1/2)2x -> base is between 0 and 1
2. a) Y values are increasing. Growth.
b) 20 % each year
c) Approx. 6 years
d) November