# 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 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
If the plane intersects the cone (either above or below) horizontally, what figure will be created?
can you not take the square root of a negative number
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
can I get some pretty basic questions
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
the solution doesn't seem right for this problem
what is the domain of f(x)=x-4/x^2-2x-15 then
x is different from -5&3
Seid
All real x except 5 and - 3
Spiro
***youtu.be/ESxOXfh2Poc
Loree
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
Don't think that you can.
Elliott
By using some imaginary no.
Tanmay
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
What are the question marks for?
Elliott
Someone should please solve it for me Add 2over ×+3 +y-4 over 5 simplify (×+a)with square root of two -×root 2 all over a multiply 1over ×-y{(×-y)(×+y)} over ×y
For the first question, I got (3y-2)/15 Second one, I got Root 2 Third one, I got 1/(y to the fourth power) I dont if it's right cause I can barely understand the question.
Is under distribute property, inverse function, algebra and addition and multiplication function; so is a combined question
Abena
find the equation of the line if m=3, and b=-2
graph the following linear equation using intercepts method. 2x+y=4
Ashley
how
Wargod
what?
John
ok, one moment
UriEl
how do I post your graph for you?
UriEl
it won't let me send an image?
UriEl
also for the first one... y=mx+b so.... y=3x-2
UriEl
y=mx+b you were already given the 'm' and 'b'. so.. y=3x-2
Tommy
Please were did you get y=mx+b from
Abena
y=mx+b is the formula of a straight line. where m = the slope & b = where the line crosses the y-axis. In this case, being that the "m" and "b", are given, all you have to do is plug them into the formula to complete the equation.
Tommy
thanks Tommy
Nimo
0=3x-2 2=3x x=3/2 then . y=3/2X-2 I think
Given
co ordinates for x x=0,(-2,0) x=1,(1,1) x=2,(2,4)
neil
"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
Where do the rays point?
Spiro
x=-b+_Гb2-(4ac) ______________ 2a
I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
so good
abdikarin
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
strategies to form the general term
carlmark
consider r(a+b) = ra + rb. The a and b are the trig identity.
Mike