# 4.1 Exponential functions  (Page 11/16)

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 $x$ 1 2 3 4 $f\left(x\right)$ 70 40 10 -20

Linear

 $x$ 1 2 3 4 $h\left(x\right)$ 70 49 34.3 24.01
 $x$ 1 2 3 4 $m\left(x\right)$ 80 61 42.9 25.61

Neither

 $x$ 1 2 3 4 $f\left(x\right)$ 10 20 40 80
 $x$ 1 2 3 4 $g\left(x\right)$ -3.25 2 7.25 12.5

Linear

For the following exercises, use the compound interest formula, $\text{\hspace{0.17em}}A\left(t\right)=P{\left(1+\frac{r}{n}\right)}^{nt}.$

After a certain number of years, the value of an investment account is represented by the equation $\text{\hspace{0.17em}}10,250{\left(1+\frac{0.04}{12}\right)}^{120}.\text{\hspace{0.17em}}$ What is the value of the account?

What was the initial deposit made to the account in the previous exercise?

$10,250$

How many years had the account from the previous exercise been accumulating interest?

An account is opened with an initial deposit of $6,500 and earns $\text{\hspace{0.17em}}3.6%\text{\hspace{0.17em}}$ interest compounded semi-annually. What will the account be worth in $\text{\hspace{0.17em}}20\text{\hspace{0.17em}}$ years? $13,268.58$ How much more would the account in the previous exercise have been worth if the interest were compounding weekly? Solve the compound interest formula for the principal, $\text{\hspace{0.17em}}P$ . $P=A\left(t\right)\cdot {\left(1+\frac{r}{n}\right)}^{-nt}$ Use the formula found in the previous exercise to calculate the initial deposit of an account that is worth $\text{\hspace{0.17em}}14,472.74\text{\hspace{0.17em}}$ after earning $\text{\hspace{0.17em}}5.5%\text{\hspace{0.17em}}$ interest compounded monthly for $\text{\hspace{0.17em}}5\text{\hspace{0.17em}}$ years. (Round to the nearest dollar.) How much more would the account in the previous two exercises be worth if it were earning interest for $\text{\hspace{0.17em}}5\text{\hspace{0.17em}}$ more years? $4,572.56$ Use properties of rational exponents to solve the compound interest formula for the interest rate, $\text{\hspace{0.17em}}r.$ Use the formula found in the previous exercise to calculate the interest rate for an account that was compounded semi-annually, had an initial deposit of$9,000 and was worth $13,373.53 after 10 years. $4%$ Use the formula found in the previous exercise to calculate the interest rate for an account that was compounded monthly, had an initial deposit of$5,500, and was worth \$38,455 after 30 years.

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

$y=3742{\left(e\right)}^{0.75t}$

continuous growth; the growth rate is greater than $\text{\hspace{0.17em}}0.$

$y=150{\left(e\right)}^{\frac{3.25}{t}}$

$y=2.25{\left(e\right)}^{-2t}$

continuous decay; the growth rate is less than $\text{\hspace{0.17em}}0.$

Suppose an investment account is opened with an initial deposit of $\text{\hspace{0.17em}}12,000\text{\hspace{0.17em}}$ earning $\text{\hspace{0.17em}}7.2%\text{\hspace{0.17em}}$ interest compounded continuously. How much will the account be worth after $\text{\hspace{0.17em}}30\text{\hspace{0.17em}}$ years?

How much less would the account from Exercise 42 be worth after $\text{\hspace{0.17em}}30\text{\hspace{0.17em}}$ years if it were compounded monthly instead?

$669.42$

## Numeric

For the following exercises, evaluate each function. Round answers to four decimal places, if necessary.

$f\left(x\right)=2{\left(5\right)}^{x},$ for $\text{\hspace{0.17em}}f\left(-3\right)$

$f\left(x\right)=-{4}^{2x+3},$ for $\text{\hspace{0.17em}}f\left(-1\right)$

$f\left(-1\right)=-4$

$f\left(x\right)={e}^{x},$ for $\text{\hspace{0.17em}}f\left(3\right)$

$f\left(x\right)=-2{e}^{x-1},$ for $\text{\hspace{0.17em}}f\left(-1\right)$

$f\left(-1\right)\approx -0.2707$

$f\left(x\right)=2.7{\left(4\right)}^{-x+1}+1.5,$ for $f\left(-2\right)$

$f\left(x\right)=1.2{e}^{2x}-0.3,$ for $\text{\hspace{0.17em}}f\left(3\right)$

$f\left(3\right)\approx 483.8146$

$f\left(x\right)=-\frac{3}{2}{\left(3\right)}^{-x}+\frac{3}{2},$ for $\text{\hspace{0.17em}}f\left(2\right)$

## Technology

For the following exercises, use a graphing calculator to find the equation of an exponential function given the points on the curve.

$\left(0,3\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(3,375\right)$

$y=3\cdot {5}^{x}$

$\left(3,222.62\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(10,77.456\right)$

$\left(20,29.495\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(150,730.89\right)$

$y\approx 18\cdot {1.025}^{x}$

$\left(5,2.909\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(13,0.005\right)$

$\left(11,310.035\right)\text{\hspace{0.17em}}$ and $\left(25,356.3652\right)$

$y\approx 0.2\cdot {1.95}^{x}$

## Extensions

The annual percentage yield (APY) of an investment account is a representation of the actual interest rate earned on a compounding account. It is based on a compounding period of one year. Show that the APY of an account that compounds monthly can be found with the formula $\text{\hspace{0.17em}}\text{APY}={\left(1+\frac{r}{12}\right)}^{12}-1.$

what is set?
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
can get some help basic precalculus
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
can get some help inverse function
ismail
Rectangle coordinate
how to find for x
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
whats a domain
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
difference between calculus and pre calculus?
give me an example of a problem so that I can practice answering
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
I want to learn about the law of exponent
explain this
what is functions?
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich