Algebra Form and Function

William G. McCallum, Eric Connally, Deborah Hughes-Hallett

Chapter 10

Exponential Functions, Expressions, and Equations - all with Video Answers

Educators

Section 1

Exponential Functions

Problem 1

Are the functions exponential? If so, identify the initial value and the growth factor.
$$
Q=12 t^{4}
$$

Nick Johnson

Nick Johnson

Numerade Educator

Problem 2

Are the functions exponential? If so, identify the initial value and the growth factor.
$$
Q=t \cdot 12^{4}
$$

Nick Johnson

Numerade Educator

Problem 3

Are the functions exponential? If so, identify the initial value and the growth factor.
$$
Q=0.75(0.2)^{t}
$$

Nick Johnson

Numerade Educator

Problem 4

Are the functions exponential? If so, identify the initial value and the growth factor.
$$
Q=0.2(3)^{0.75 t}
$$

Nick Johnson

Numerade Educator

Problem 5

Write the exponential functions in the form $Q=a b^{t}$ and identify the initial value and growth factor.
$$
Q=300 \cdot 3^{t}
$$

Nick Johnson

Numerade Educator

Problem 6

Write the exponential functions in the form $Q=a b^{t}$ and identify the initial value and growth factor.
$$
Q=\frac{190}{3^{t}}
$$

Nick Johnson

Numerade Educator

Problem 7

Write the exponential functions in the form $Q=a b^{t}$ and identify the initial value and growth factor.
$$
Q=200 \cdot 3^{2 t}
$$

Nick Johnson

Numerade Educator

Problem 8

Write the exponential functions in the form $Q=a b^{t}$ and identify the initial value and growth factor.
$$
Q=50 \cdot 2^{-t}
$$

Nick Johnson

Numerade Educator

Problem 9

Can the quantities be represented by exponential functions? Explain.
The price of gas if it grows by $\$ 0.02$ a week.

Nick Johnson

Numerade Educator

Problem 10

Can the quantities be represented by exponential functions? Explain.
The quantity of a prescribed drug in the bloodstream if it shrinks by a factor of 0.915 every 4 hours.

Nick Johnson

Numerade Educator

Problem 11

Can the quantities be represented by exponential functions? Explain.
The speed of personal computers if it doubles every 3 years.

Nick Johnson

Numerade Educator

Problem 12

Can the quantities be represented by exponential functions? Explain.
The height of a baseball thrown straight up into the air and then caught.

Nick Johnson

Numerade Educator

Problem 13

The population of a town is $M$ in 2005 and is growing with a yearly growth factor of $Z .$ Write a formula that gives the population, $P$, at time $t$ years after 2005 .

Nick Johnson

Numerade Educator

Problem 14

Write expressions representing the quantities described.
The balance $B$ increases $n$ times in a row by a factor of 1.11

Nick Johnson

Numerade Educator

Problem 15

Write expressions representing the quantities described.
The population is initially 800 and grows by a factor of $d$ a total of $k$ times in a row.

Nick Johnson

Numerade Educator

Problem 16

Write expressions representing the quantities described.
The population starts at 2000 and increases by a factor of $k$ five times in a row.

Nick Johnson

Numerade Educator

Problem 17

Write expressions representing the quantities described.
The investment value $V_{0}$ drops by a third $n$ times in a
row.

Nick Johnson

Numerade Educator

Problem 18

Write expressions representing the quantities described.
The area covered by marsh starts at $R$ acres and decreases by a factor of $s$ nine times in a row.

Nick Johnson

Numerade Educator

Problem 19

Write expressions representing the quantities described.
The number infected $N$ increases by a factor of $z$ for $t$ times in a row.

Nick Johnson

Numerade Educator

Problem 20

A lump of uranium is decaying exponentially with time, $t$. Which of graphs (I)-(IV) could represent this?

Nick Johnson

Numerade Educator

Problem 21

Which of graphs (I)-(IV) might be exponential?

Nick Johnson

Numerade Educator

Problem 22

An investment initially worth $\$ 3000$ grows by a factor of 1.062 every year. Complete Table 10.1 , which shows the value of the investment over time.

Alison Rodriguez

Alison Rodriguez

Numerade Educator

Problem 23

Match the exponential functions (a)-(d) with their graphs (I)-(IV).
(a) $Q=4(1.2)^{t}$
(b) $Q=4(0.7)^{t}$
(c) $Q=8(1.2)^{t}$
(d) $Q=8(1.4)^{t}$

Ashima Tiwari

Numerade Educator

Problem 24

Sketch graphs of the exponential functions. Label your axes.
$$
y=100(1.04)^{t}
$$

Ashima Tiwari

Numerade Educator

Problem 25

Sketch graphs of the exponential functions. Label your axes.
$$
y=800(0.9)^{t}
$$

Swati Agarwal

Numerade Educator

Problem 26

In $2005,$ Bhutan had a population $^{1}$ of about 2,200,000 and an annual growth factor of 1.0211 . Let $f(t)$ be the population $t$ years after 2005 assuming growth continues at this rate.
(a) Evaluate the following expressions and explain what they tell you about Bhutan.
(i) $f(1)$
(ii) $f(2)$
(iii) $f(3)$
(iv) $f(5)$
(b) Write a formula for $f(t)$.

Yujie Wang

College of San Mateo

Problem 27

In $2005,$ the Czech Republic had a population of about 10,200,000 and a growth factor of 0.9995 (that is, the population was shrinking) ${ }^{2}$ Let $f(t)$ be the population $t$ years after 2005 assuming growth continues at this rate.
(a) Evaluate the following expressions and explain what they tell you about the Czech Republic.
(i) $f(1)$
(ii) $f(2)$
(iii) $f(3)$
(iv) $f(5)$
(b) Write a formula for $f(t)$.

Lily An

Numerade Educator

Problem 28

Write an expression that represents the population of a bacteria colony that starts with 10,000 members and
(a) Halves twice
(b) Is multiplied 8 times by 0.7
(c) Is multiplied 4 times by 1.5

Nick Johnson

Numerade Educator

Problem 29

The population of a city after $t$ years is given by $220,000(1.016)^{t}$. Identify the initial value and the growth factor and explain what they mean in terms of the city.

Nick Johnson

Numerade Educator

Problem 30

The number of grams of palladium-98 remaining after $\underline{t}$ minutes is given by
$$
Q=600(0.962)^{t}
$$
Find the quantity remaining after
(a) 1 minute
(b) 2 minutes
(c) 1 hour

Carson Merrill

Numerade Educator

Problem 31

An investment grows by 1.2 times in the first year, by 1.3 times in the second year, then drops by a factor of 0.88 in the third year. By what overall factor does the value change over the three-year period?

Alison Rodriguez

Alison Rodriguez

Numerade Educator

Problem 32

The value $V$, in dollars, of an investment after $t$ years is given by the function
$$
V=g(t)=3000(1.08)^{t}
$$
Plot the value of the investment at 5 -year intervals over a 30 -year period beginning with $t=0 .$ By how much does the investment grow during the first ten years? The second ten years? The third ten years?

Melissa Munoz

Numerade Educator

Problem 33

A mill in a town with a population of 400 closes and people begin to move away. Find a possible formula for the number of people $P$ in year $t$ if each year one-fifth of the remaining population leaves.

Amrita Bhasin

Numerade Educator

Problem 34

Write the exponential functions in the form $Q=a b^{t},$ and identify the initial value and the growth factor.
$$
Q=\frac{50}{2^{t / 12}}
$$

Nick Johnson

Numerade Educator

Problem 35

Write the exponential functions in the form $Q=a b^{t},$ and identify the initial value and the growth factor.
$$
Q=250 \cdot 5^{-2 t-1}
$$

Nick Johnson

Numerade Educator

Problem 36

Write the exponential functions in the form $Q=a b^{t},$ and identify the initial value and the growth factor.
$$
Q=2^{t} \sqrt{300}
$$

Nick Johnson

Numerade Educator

Problem 37

Write the exponential functions in the form $Q=a b^{t},$ and identify the initial value and the growth factor.
$$
Q=40 \cdot 2^{2 t}
$$

Nick Johnson

Numerade Educator

Problem 38

Write the exponential functions in the form $Q=a b^{t},$ and identify the initial value and the growth factor.
$$
Q=45 \cdot 3^{t-2}
$$

Nick Johnson

Numerade Educator

Problem 39

Write the exponential functions in the form $Q=a b^{t},$ and identify the initial value and the growth factor.
$$
Q=\frac{\sqrt{100^{t}}}{5}
$$

Nick Johnson

Numerade Educator

Problem 40

Do the following expressions define exponential functions? If so, identify the values of $a$ and $b$.
(a) $3 \cdot 0.5^{x}$
(b) $8^{t}$
(c) $8^{1-2 x}$

James Kiss

Numerade Educator

Problem 41

Put the function in the required form and state the values of all constants.
$$
y=3 x+8 \text { in the form } y=5+m\left(x-x_{0}\right) \text { . }
$$

Lauren Shelton

Numerade Educator

Problem 42

Put the function in the required form and state the values of all constants.
$$
y=3(x \sqrt{7})^{3} \text { in the form } y=k x^{p} .
$$

Victor Salazar

Numerade Educator

Problem 43

Put the function in the required form and state the values of all constants.
$y=3 x^{3}-2 x^{2}+4 x+5$ in the form
$$
y=a+x(b+x(c+d x)) \text { . }
$$

Charles Carter

Numerade Educator

Problem 44

Put the function in the required form and state the values of all constants.
$$
y=\frac{(\sqrt{3})^{4 t}}{5} \text { in the form } y=a b^{t} \text { . }
$$

Alison Rodriguez

Alison Rodriguez

Numerade Educator