Math 121.
Exponential Functions
Fall 2016
Instructions. Work in groups of 3 to solve the following problems. Turn them in at the end of class
for credit.
Names.
1. Use properties of exponents to solve 84x−7 = 87
x+7
(calculators should not be used).
Solution: Since 8 > 0, and 8 6= 1, these expressions are defined, and the exponents must be
equal, and so
4x − 7 = 7(x + 7) ⇒ 4x − 7 = 7x + (7)(7) ⇒ (−3)(x) = 56 ⇒ x =
56
−3
where the reader can simplify the final fraction when possible.
2. Consider the function f (x) = 6x
(a) Complete the following table of values for f .
−3
x
−2
−1
0
1
2
3
y
(b) Sketch a graph of f , and on the same coordinate axes sketch y = x and the graph of f −1 (x)
(c) Using your answer from (b), sketch g(x) = 6x+3 − 4.
Solution: (a) The completed table is as follows.
x
−3
−2
−1
0
1
2
3
y
1/216
1/36
1/6
1
6
36
216
In (b), recall that the graph of the inverse will be the graph of f reflected over the line y = x. In
(c), the graph of y = 6x is shifted 3 units to the left and 4 units down to obtain the graph of g.
(b)
(c)
8 y
8 y
6
6
4
f
4
f −1
2
2
x
−8 −6 −4 −2
−2
x
−4
−8 −6 −4 −2
−2
g
−4
−6
−6
−8
−8
2
4
6
8
2
4
6
8
3. The graph of an exponential function y = bx is given on the graph below.
8 y
6
4
2
x
−8 −6 −4 −2
−2
2
4
6
8
−4
−6
−8
(a) Use the graph to estimate b.
(b) Using the graph from (a), graph y = b|x| .
(c) Using your graph from (b), graph y = b|x+3| − 5.
Solution: (a) b =
1
5
because the graph goes through the point (−1, 5).
(b) This graph results in reflecting the part of the original graph for x ≥ 0 over the y-axis.
(c) The requested graph results from shifting the original graph 3 units to the left and 5 units
down. The graph is given below
8 y
6
4
2
−8 −6 −4 −2
−2
(c)
(b) x
2
4
−4
−6
−8
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6
8
4. The number of bass in a lake is given by
P (t) =
3420
1 + 5e−0.06t
where t is the number of months that have passed since the lake was stocked with bass.
(a) How many bass were in the lake immediately after it was stocked?
(b) How many bass were in the lake 2 years after it was stocked? Round your answer to the nearest
whole number.
(c) What will happen to the bass population as t increases without bound?
Solution: (a) P (0) =
3420
1+5
= 570 bass were in the lake immediately after it was stocked.
3420
≈ 1565.48 ≈ 1565 bass were in the lake 2 years after it was stocked.
1 + 5e(−0.06)(24)
(c) e−0.06t is an exponential function with base b = e−0.06 < 1, this will approach 0 as t increases
without bound. Therefore, the bass population will increase, approaching 3420 as time increases
without bound.
(b) P (24) =
5. The population of a small city is currently 80000 and is growing at 3 percent per year. Thus the
population is given by
P (t) = 80000(1.03)t
where t is time measured in years from the present.
(a) What will the population of the city be in one year?
(b) According to this model, what will the population of the city be in 14 years from now? Express
answer to the nearest whole number.
(c) Suppose Charles has an investment account that is growing a a rate of 3 percent per year, and he
currently has 80000 dollars in the account. How much money will be in the account 14 years from
now? Express answer to the nearest dollar.
Solution: (a) P (1) = 80000(1.03) = 82400 which represents an increase of 3 percent over the
current populaton.
(b) Assuming the current growth rate continues, the population will be
P (t) = 80000(1.03)14 = 121007.
(c) This is mathematically the same question as (b), except expressed in dollars, not population,
so the account will have 80000(1.03)14 = 121007 dollars in it after 14 years.
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6. Exponential functions are ideal for dealing with large numbers (or extremely small numbers),
because once we know the base, only the exponent changes. For example 106781.2346 and 10781.8457
are numbers larger than many calculators will accept, yet properties of exponents make them easy to
multiply (or divide). Indeed,
106781.2346 × 10781.8457 = 106781.2346+781.8457 = 107563.0803
Properties of exponents also make it easy to write the above answer in scientific notation
107563.0803 = 107563+.0803 = 100.0803 × 107563 ≈ 1.2031 × 107563
Use properties of exponents to answer the following questions.
(a) Write 101564.858 in scientific notation. Use 4 significant figures in your final answer.
(b) Find the product 101564.858 · 10659.340 as a power of 10.
(c) Convert your answer in (b) to scientific notation. Use 4 significant digits in your answer.
Solution: (a) 101564.858 = 101564+0.858 = 100.858 × 101564 ≈ 7.211 × 101564
(b) 101564.858 · 10659.340 = 101564.858+659.340 = 102224.198
(c) 102224.198 = 102224+0.198 = 100.198 × 102224 ≈ 1.578 × 102224
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