# 6.1 Exponential functions  (Page 6/16)

 Page 6 / 16

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.

what are you up to?
nothing up todat yet
Miranda
hi
jai
hello
jai
Miranda Drice
jai
aap konsi country se ho
jai
which language is that
Miranda
I am living in india
jai
good
Miranda
what is the formula for calculating algebraic
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
hi
jai
hi Miranda
jai
thanks
Propessor
welcome
jai
What is algebra
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math. I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
lolll who told you I'm good at it
Miranda
something seems to wispher me to my ear that u are good at it. lol
Jeffrey
lolllll if you say so
Miranda
but seriously, Im really bad at math. And I hate it. But you see, I downloaded this app two months ago hoping to master it.
Jeffrey
which grade are you in though
Miranda
oh woww I understand
Miranda
Jeffrey
Jeffrey
Miranda
how come you finished in college and you don't like math though
Miranda
gotta practice, holmie
Steve
if you never use it you won't be able to appreciate it
Steve
I don't know why. But Im trying to like it.
Jeffrey
yes steve. you're right
Jeffrey
so you better
Miranda
what is the solution of the given equation?
which equation
Miranda
I dont know. lol
Jeffrey
Miranda
Jeffrey
answer and questions in exercise 11.2 sums
how do u calculate inequality of irrational number?
Alaba
give me an example
Chris
and I will walk you through it
Chris
cos (-z)= cos z .
cos(- z)=cos z
Mustafa
what is a algebra
(x+x)3=?
6x
Obed
what is the identity of 1-cos²5x equal to?
__john __05
Kishu
Hi
Abdel
hi
Ye
hi
Nokwanda
C'est comment
Abdel
Hi
Amanda
hello
SORIE
Hiiii
Chinni
hello
Ranjay
hi
ANSHU
hiiii
Chinni
h r u friends
Chinni
yes
Hassan
so is their any Genius in mathematics here let chat guys and get to know each other's
SORIE
I speak French
Abdel
okay no problem since we gather here and get to know each other
SORIE
hi im stupid at math and just wanna join here
Yaona
lol nahhh none of us here are stupid it's just that we have Fast, Medium, and slow learner bro but we all going to work things out together
SORIE
it's 12
what is the function of sine with respect of cosine , graphically
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
sinx sin2x is linearly dependent
what is a reciprocal
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
I don't understand how radicals works pls
How look for the general solution of a trig function