# 6.1 Exponential functions  (Page 10/16)

 Page 10 / 16

## Algebraic

For the following exercises, identify whether the statement represents an exponential function. Explain.

The average annual population increase of a pack of wolves is 25.

A population of bacteria decreases by a factor of $\text{\hspace{0.17em}}\frac{1}{8}\text{\hspace{0.17em}}$ every $\text{\hspace{0.17em}}24\text{\hspace{0.17em}}$ hours.

exponential; the population decreases by a proportional rate. .

The value of a coin collection has increased by $\text{\hspace{0.17em}}3.25%\text{\hspace{0.17em}}$ annually over the last $\text{\hspace{0.17em}}20\text{\hspace{0.17em}}$ years.

For each training session, a personal trainer charges his clients $\text{\hspace{0.17em}}\text{}5\text{\hspace{0.17em}}$ less than the previous training session.

not exponential; the charge decreases by a constant amount each visit, so the statement represents a linear function. .

The height of a projectile at time $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is represented by the function $\text{\hspace{0.17em}}h\left(t\right)=-4.9{t}^{2}+18t+40.$

For the following exercises, consider this scenario: For each year $\text{\hspace{0.17em}}t,$ the population of a forest of trees is represented by the function $\text{\hspace{0.17em}}A\left(t\right)=115{\left(1.025\right)}^{t}.\text{\hspace{0.17em}}$ In a neighboring forest, the population of the same type of tree is represented by the function $\text{\hspace{0.17em}}B\left(t\right)=82{\left(1.029\right)}^{t}.\text{\hspace{0.17em}}$ (Round answers to the nearest whole number.)

Which forest’s population is growing at a faster rate?

The forest represented by the function $\text{\hspace{0.17em}}B\left(t\right)=82{\left(1.029\right)}^{t}.$

Which forest had a greater number of trees initially? By how many?

Assuming the population growth models continue to represent the growth of the forests, which forest will have a greater number of trees after $\text{\hspace{0.17em}}20\text{\hspace{0.17em}}$ years? By how many?

After $\text{\hspace{0.17em}}t=20\text{\hspace{0.17em}}$ years, forest A will have $\text{\hspace{0.17em}}43\text{\hspace{0.17em}}$ more trees than forest B.

Assuming the population growth models continue to represent the growth of the forests, which forest will have a greater number of trees after $\text{\hspace{0.17em}}100\text{\hspace{0.17em}}$ years? By how many?

Discuss the above results from the previous four exercises. Assuming the population growth models continue to represent the growth of the forests, which forest will have the greater number of trees in the long run? Why? What are some factors that might influence the long-term validity of the exponential growth model?

Answers will vary. Sample response: For a number of years, the population of forest A will increasingly exceed forest B, but because forest B actually grows at a faster rate, the population will eventually become larger than forest A and will remain that way as long as the population growth models hold. Some factors that might influence the long-term validity of the exponential growth model are drought, an epidemic that culls the population, and other environmental and biological factors.

For the following exercises, determine whether the equation represents exponential growth, exponential decay, or neither. Explain.

$y=300{\left(1-t\right)}^{5}$

$y=220{\left(1.06\right)}^{x}$

exponential growth; The growth factor, $\text{\hspace{0.17em}}1.06,$ is greater than $\text{\hspace{0.17em}}1.$

$y=16.5{\left(1.025\right)}^{\frac{1}{x}}$

$y=11,701{\left(0.97\right)}^{t}$

exponential decay; The decay factor, $\text{\hspace{0.17em}}0.97,$ is between $\text{\hspace{0.17em}}0\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}1.$

For the following exercises, find the formula for an exponential function that passes through the two points given.

$\left(0,6\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(3,750\right)$

$\left(0,2000\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(2,20\right)$

$f\left(x\right)=2000{\left(0.1\right)}^{x}$

$\left(-1,\frac{3}{2}\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(3,24\right)$

$\left(-2,6\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(3,1\right)$

$f\left(x\right)={\left(\frac{1}{6}\right)}^{-\frac{3}{5}}{\left(\frac{1}{6}\right)}^{\frac{x}{5}}\approx 2.93{\left(0.699\right)}^{x}$

$\left(3,1\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(5,4\right)$

For the following exercises, determine whether the table could represent a function that is linear, exponential, or neither. If it appears to be exponential, find a function that passes through the points.

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
The sequence is {1,-1,1-1.....} has
how can we solve this problem
Sin(A+B) = sinBcosA+cosBsinA
Prove it
Eseka
Eseka
hi
Joel
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
7.5 and 37.5
Nando
find the sum of 28th term of the AP 3+10+17+---------
I think you should say "28 terms" instead of "28th term"
Vedant
the 28th term is 175
Nando
192
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