# 4.1 Exponential functions  (Page 6/16)

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Find an equation for the exponential function graphed in [link] .

$f\left(x\right)=\sqrt{2}{\left(\sqrt{2}\right)}^{x}.\text{\hspace{0.17em}}$ Answers may vary due to round-off error. The answer should be very close to $\text{\hspace{0.17em}}1.4142{\left(1.4142\right)}^{x}.$

Given two points on the curve of an exponential function, use a graphing calculator to find the equation.

1. Press [STAT].
2. Clear any existing entries in columns L1 or L2.
3. In L1 , enter the x -coordinates given.
4. In L2 , enter the corresponding y -coordinates.
5. Press [STAT] again. Cursor right to CALC , scroll down to ExpReg (Exponential Regression) , and press [ENTER].
6. The screen displays the values of a and b in the exponential equation $\text{\hspace{0.17em}}y=a\cdot {b}^{x}$ .

## Using a graphing calculator to find an exponential function

Use a graphing calculator to find the exponential equation that includes the points $\text{\hspace{0.17em}}\left(2,24.8\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(5,198.4\right).$

Follow the guidelines above. First press [STAT] , [EDIT] , [1: Edit…], and clear the lists L1 and L2 . Next, in the L1 column, enter the x -coordinates, 2 and 5. Do the same in the L2 column for the y -coordinates, 24.8 and 198.4.

Now press [STAT] , [CALC] , [0: ExpReg] and press [ENTER] . The values $\text{\hspace{0.17em}}a=6.2\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}b=2\text{\hspace{0.17em}}$ will be displayed. The exponential equation is $\text{\hspace{0.17em}}y=6.2\cdot {2}^{x}.$

Use a graphing calculator to find the exponential equation that includes the points (3, 75.98) and (6, 481.07).

$y\approx 12\cdot {1.85}^{x}$

## Applying the compound-interest formula

Savings instruments in which earnings are continually reinvested, such as mutual funds and retirement accounts, use compound interest    . The term compounding refers to interest earned not only on the original value, but on the accumulated value of the account.

The annual percentage rate (APR)    of an account, also called the nominal rate    , is the yearly interest rate earned by an investment account. The term  nominal  is used when the compounding occurs a number of times other than once per year. In fact, when interest is compounded more than once a year, the effective interest rate ends up being greater than the nominal rate! This is a powerful tool for investing.

We can calculate the compound interest using the compound interest formula, which is an exponential function of the variables time $\text{\hspace{0.17em}}t,$ principal $\text{\hspace{0.17em}}P,$ APR $\text{\hspace{0.17em}}r,$ and number of compounding periods in a year $\text{\hspace{0.17em}}n:$

$A\left(t\right)=P{\left(1+\frac{r}{n}\right)}^{nt}$

For example, observe [link] , which shows the result of investing $1,000 at 10% for one year. Notice how the value of the account increases as the compounding frequency increases. Frequency Value after 1 year Annually$1100
Semiannually $1102.50 Quarterly$1103.81
Monthly $1104.71 Daily$1105.16

## The compound interest formula

Compound interest can be calculated using the formula

$A\left(t\right)=P{\left(1+\frac{r}{n}\right)}^{nt}$

where

• $A\left(t\right)\text{\hspace{0.17em}}$ is the account value,
• $t\text{\hspace{0.17em}}$ is measured in years,
• $P\text{\hspace{0.17em}}$ is the starting amount of the account, often called the principal, or more generally present value,
• $r\text{\hspace{0.17em}}$ is the annual percentage rate (APR) expressed as a decimal, and
• $n\text{\hspace{0.17em}}$ is the number of compounding periods in one year.

## Calculating compound interest

If we invest $3,000 in an investment account paying 3% interest compounded quarterly, how much will the account be worth in 10 years? Because we are starting with$3,000, $\text{\hspace{0.17em}}P=3000.\text{\hspace{0.17em}}$ Our interest rate is 3%, so Because we are compounding quarterly, we are compounding 4 times per year, so $\text{\hspace{0.17em}}n=4.\text{\hspace{0.17em}}$ We want to know the value of the account in 10 years, so we are looking for $\text{\hspace{0.17em}}A\left(10\right),$ the value when

The account will be worth about $4,045.05 in 10 years. #### Questions & Answers The average annual population increase of a pack of wolves is 25. Brittany Reply how do you find the period of a sine graph Imani Reply Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period Am if not then how would I find it from a graph Imani by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates. Am you could also do it with two consecutive minimum points or x-intercepts Am I will try that thank u Imani Case of Equilateral Hyperbola Jhon Reply ok Zander ok Shella f(x)=4x+2, find f(3) Benetta f(3)=4(3)+2 f(3)=14 lamoussa 14 Vedant pre calc teacher: "Plug in Plug in...smell's good" f(x)=14 Devante 8x=40 Chris Explain why log a x is not defined for a < 0 Baptiste Reply the sum of any two linear polynomial is what Esther Reply divide simplify each answer 3/2÷5/4 Momo Reply divide simplify each answer 25/3÷5/12 Momo how can are find the domain and range of a relations austin Reply the range is twice of the natural number which is the domain Morolake A cell phone company offers two plans for minutes. Plan A:$15 per month and $2 for every 300 texts. Plan B:$25 per month and \$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
6000
Robert
more than 6000
Robert
can I see the picture
How would you find if a radical function is one to one?
how to understand calculus?
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
what is foci?
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations