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Evaluate and Graph Exponential Functions

Graph exponential functions

Graph Exponential Functions

The functions we have studied so far do not give us a model for many naturally occurring phenomena. From the growth of populations and the spread of viruses to radioactive decay and compounding interest, the models are very different from what we have studied so far. These models involve exponential functions.

An exponential function is a function of the form $f(x)={a}^{x}$ where $a>0$ and $a\ne 1.$

Notice that in this function, the variable is the exponent. In our functions so far, the variables were the base.

Our definition says $a\ne 1.$ If we let $a=1,$ then $f(x)={a}^{x}$ becomes $f(x)={1}^{x}.$ Since ${1}^{x}=1$ for all real numbers, $f(x)={1}^{}.$ This is the constant function.

Our definition also says $a>0.$ If we let a base be negative, say $-4,$ then $f(x)={(-4)}^{x}$ is not a real number when $x=\frac{1}{2}.$

\[\begin{array}{lll}f(x) & = & {(-4)}^{x} \\ f(\frac{1}{2}) & = & {(-4)}^{\frac{1}{2}} \\ f(\frac{1}{2}) & = & \sqrt{-4}\ \text{not a real number}\end{array}\]

In fact, $f(x)={(-4)}^{x}$ would not be a real number any time $x$ is a fraction with an even denominator. So our definition requires $a>0.$

By graphing a few exponential functions, we will be able to see their unique properties.

Example

Try it.

On the same coordinate system graph $f(x)={2}^{x}$ and $g(x)={3}^{x}.$

Solution

We will use point plotting to graph the functions.

Example

Try it.

On the same coordinate system, graph $f(x)={(\frac{1}{2})}^{x}$ and $g(x)={(\frac{1}{3})}^{x}.$

Solution

We will use point plotting to graph the functions.

Example

Try it.

On the same coordinate system graph $f(x)={2}^{x}$ and $g(x)={2}^{x+1}.$

Solution

We will use point plotting to graph the functions.

\[e\approx 2.718281828\]

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Solve Exponential Equations

Equations that include an exponential expression ${a}^{x}$ are called exponential equations. To solve them we use a property that says as long as $a>0$ and $a\ne 1,$ if ${a}^{x}={a}^{y}$ then it is true that $x=y.$ In other words, in an exponential equation, if the bases are equal then the exponents are equal.

To use this property, we must be certain that both sides of the equation are written with the same base.

How to Solve an Exponential Equation

Try it.

Solve: ${3}^{2x-5}=27.$

Solution

The steps are summarized below.

In the next example, we will use our properties on exponents.

Example

Try it.

Solve $\frac{{e}^{{x}^{2}}}{{e}^{3}}={e}^{2x}$.

Solution

	$\ \frac{{e}^{{x}^{2}}}{{e}^{3}}={e}^{2x}$
Use the Property of Exponents: $\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}.$	$\ {e}^{{x}^{2}-3}={e}^{2x}$
Write a new equation by setting the exponents equal.	$\ {x}^{2}-3=2x\$
Solve the equation.	${x}^{2}-2x-3=0\$
	$(x-3)(x+1)=0\$
	$x=3,\ x=-1$
Check the solutions.

Use Exponential Models in Applications

Exponential functions model many situations. If you own a bank account, you have experienced the use of an exponential function. There are two formulas that are used to determine the balance in the account when interest is earned. If a principal, P, is invested at an interest rate, r, for t years, the new balance, A, will depend on how often the interest is compounded. If the interest is compounded n times a year we use the formula $A=P{(1+\frac{r}{n})}^{nt}.$ If the interest is compounded continuously, we use the formula $A=P{e}^{rt}.$ These are the formulas for compound interest.

As you work with the Interest formulas, it is often helpful to identify the values of the variables first and then substitute them into the formula.

Example

Try it.

A total of $\text{\$}10,000$ was invested in a college fund for a new grandchild. If the interest rate is $5\text{\%},$ how much will be in the account in 18 years by each method of compounding?

ⓐ compound quarterly

ⓑ compound monthly

Solution

	$\ A=?$
Identify the values of each variable in the formulas.	$\ P=\text{\$}10,000$
Remember to express the percent as a decimal.	$\ r=0.05$
	$\ t=18\ \text{years}$

ⓐ

For quarterly compounding, $n=4$. There are 4 quarters in a year.	$\ A=P{(1+\frac{r}{n})}^{nt}$
Substitute the values in the formula.	$\ A=10,000{(1+\frac{0.05}{4})}^{4\cdot 18}$
Compute the amount. Be careful to consider the order of operations as you enter the expression into your calculator.	$\ A=\text{\$}24,459.20$

ⓑ

For monthly compounding, $n=12$. There are 12 months in a year.	$\ A=P{(1+\frac{r}{n})}^{nt}$
Substitute the values in the formula.	$\ A=10,000{(1+\frac{0.05}{12})}^{12\cdot 18}$
Compute the amount.	$\ A=\text{\$}24,550.08$

ⓒ

For compounding continuously,	$\ A=P{e}^{rt}$
Substitute the values in the formula.	$\ A=10,000{e}^{0.05\cdot 18}$
Compute the amount.	$\ A=\text{\$}24,596.03$

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Key Concepts

Properties of the Graph of $f(x)={a}^{x}:$

when $a>1$	when \(0
Domain	$(\text{-}\infty ,\infty )$	Domain	$(\text{-}\infty ,\infty )$
Range	$(0,\infty )$	Range	$(0,\infty )$
$x$-intercept	none	$x$-intercept	none
$y$-intercept	$(0,1)$	$y$-intercept	$(0,1)$
Contains	$(1,a),\ (-1,\frac{1}{a})$	Contains	$(1,a),\ (-1,\frac{1}{a})$
Asymptote	$x$-axis, the line $y=0$	Asymptote	$x$-axis, the line $y=0$
Basic shape	increasing	Basic shape	decreasing

One-to-One Property of Exponential Functions:
For $a>0$ and $a\ne 1,$
\[\text{If}\ {a}^{x}={a}^{y},\ \text{then}\ x=y.\]
How to Solve an Exponential Equation
1. Write both sides of the equation with the same base, if possible.
2. Write a new equation by setting the exponents equal.
3. Solve the equation.
4. Check the solution.
Compound Interest: For a principal, $P,$ invested at an interest rate, $r,$ for $t$ years, the new balance, $A,$ is
$\begin{array}{llllll} \\ \\ A=P{(1+\frac{r}{n})}^{nt} & & & & & \text{when compounded}\ n\ \text{times a year.} \\ A=P{e}^{rt} & & & & & \text{when compounded continuously.}\end{array}$
Exponential Growth and Decay: For an original amount, ${A}_{0}$ that grows or decays at a rate, $r,$ for a certain time $t,$ the final amount,$A,$ is $A={A}_{0}{e}^{rt}.$

Evaluate and Graph Exponential Functions

Graph Exponential Functions

In the following exercises, graph each exponential function.

Try it.

$f(x)={2}^{x}$

Solution

Try it.

$g(x)={3}^{x}$

Try it.

$f(x)={6}^{x}$

Solution

Try it.

$g(x)={7}^{x}$

Try it.

$f(x)={(1.5)}^{x}$

Solution

Try it.

$g(x)={(2.5)}^{x}$

Try it.

$f(x)={(\frac{1}{2})}^{x}$

Solution

Try it.

$g(x)={(\frac{1}{3})}^{x}$

Try it.

$f(x)={(\frac{1}{6})}^{x}$

Solution

Try it.

$g(x)={(\frac{1}{7})}^{x}$

Try it.

$f(x)={(0.4)}^{x}$

Solution

Try it.

$g(x)={(0.6)}^{x}$

In the following exercises, graph each function in the same coordinate system.

Try it.

$f(x)={4}^{x},\ g(x)={4}^{x-1}$

Solution

Try it.

$f(x)={3}^{x},\ g(x)={3}^{x-1}$

Try it.

$f(x)={2}^{x},\ g(x)={2}^{x-2}$

Solution

Try it.

$f(x)={2}^{x},\ g(x)={2}^{x+2}$

Try it.

$f(x)={3}^{x},\ g(x)={3}^{x}+2$

Solution

Try it.

$f(x)={4}^{x},\ g(x)={4}^{x}+2$

Try it.

$f(x)={2}^{x},\ g(x)={2}^{x}+1$

Solution

Try it.

$f(x)={2}^{x},\ g(x)={2}^{x}-1$

In the following exercises, graph each exponential function.

Try it.

$f(x)={3}^{x+2}$

Solution

Try it.

$f(x)={3}^{x-2}$

Try it.

$f(x)={2}^{x}+3$

Solution

Try it.

$f(x)={2}^{x}-3$

Try it.

$f(x)={(\frac{1}{2})}^{x-4}$

Solution

Try it.

$f(x)={(\frac{1}{2})}^{x}-3$

Try it.

$f(x)={e}^{x}+1$

Solution

Try it.

$f(x)={e}^{x-2}$

Try it.

$f(x)=\text{-}{2}^{x}$

Solution

Try it.

$f(x)={2}^{-x-1}-1$

Solve Exponential Equations

In the following exercises, solve each equation.

Try it.

${2}^{3x-8}=16$

Solution

$x=4$

Try it.

${2}^{2x-3}=32$

Try it.

${3}^{x+3}=9$

Solution

$x=-1$

Try it.

${3}^{{x}^{2}}=81$

Try it.

${4}^{{x}^{2}}=4$

Solution

$x=-1,x=1$

Try it.

${4}^{x}=32$

Try it.

${4}^{x+2}=64$

Solution

$x=1$

Try it.

${4}^{x+3}=16$

Try it.

${2}^{{x}^{2}+2x}=\frac{1}{2}$

Solution

$x=-1$

Try it.

${3}^{{x}^{2}-2x}=\frac{1}{3}$

Try it.

${e}^{3x}\cdot {e}^{4}={e}^{10}$

Solution

$x=2$

Try it.

${e}^{2x}\cdot {e}^{3}={e}^{9}$

Try it.

$\frac{{e}^{{x}^{2}}}{{e}^{2}}={e}^{x}$

Solution

$x=-1,x=2$

Try it.

$\frac{{e}^{{x}^{2}}}{{e}^{3}}={e}^{2x}$

In the following exercises, match the graphs to one of the following functions: ⓐ ${2}^{x}$ ⓑ ${2}^{x+1}$ ⓒ ${2}^{x-1}$ ⓓ ${2}^{x}+2$ ⓔ ${2}^{x}-2$ ⓕ ${3}^{x}$

Try it.

Solution

ⓕ

Try it.

Solution

ⓐ

Try it.

Solution

ⓔ

Try it.

Use exponential models in applications

In the following exercises, use an exponential model to solve.

Try it.

Edgar accumulated $\text{\$}5,000$ in credit card debt. If the interest rate is $20\text{\%}$ per year, and he does not make any payments for 2 years, how much will he owe on this debt in 2 years by each method of compounding? ⓐ compound quarterly ⓑ compound monthly ⓒ compound continuously

Solution

Try it.

Cynthia invested $\text{\$}12,000$ in a savings account. If the interest rate is $6\text{\%},$ how much will be in the account in 10 years by each method of compounding? ⓐ compound quarterly
ⓑ compound monthly ⓒ compound continuously

Try it.

Rochelle deposits $\text{\$}5,000$ in an IRA. What will be the value of her investment in 25 years if the investment is earning $8\text{\%}$ per year and is compounded continuously?

Solution

$\text{\$}36,945.28$

Try it.

Nazerhy deposits $\text{\$}8,000$ in a certificate of deposit. The annual interest rate is $6\text{\%}$ and the interest will be compounded quarterly. How much will the certificate be worth in 10 years?

Try it.

A researcher at the Center for Disease Control and Prevention is studying the growth of a bacteria. He starts his experiment with 100 of the bacteria that grows at a rate of $6\text{\%}$ per hour. He will check on the bacteria every 8 hours. How many bacteria will he find in 8 hours?

Solution

162 bacteria

Try it.

A biologist is observing the growth pattern of a virus. She starts with 50 of the virus that grows at a rate of $20\text{\%}$ per hour. She will check on the virus in 24 hours. How many viruses will she find?

Try it.

In the last ten years the population of Indonesia has grown at a rate of $1.12\text{\%}$ per year to 258,316,051. If this rate continues, what will be the population in 10 more years?

Solution

288,929,825

Try it.

In the last ten years the population of Brazil has grown at a rate of $0.9\text{\%}$ per year to 205,823,665. If this rate continues, what will be the population in 10 more years?

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Simplify: $(\frac{{x}^{3}}{{x}^{2}}).$
If you missed this problem, review .

விடை தெரியப்படுத்து

$x$
Evaluate: ⓐ ${2}^{0}$ ⓑ ${(\frac{1}{3})}^{0}.$
If you missed this problem, review .

விடை தெரியப்படுத்து

ⓐ $1$; ⓑ $1$
Evaluate: ⓐ ${2}^{-1}$ ⓑ ${(\frac{1}{3})}^{-1}.$
If you missed this problem, review .

விடை தெரியப்படுத்து

ⓐ $\frac{1}{2}$; ⓑ $3$
On the same coordinate system graph $f(x)={2}^{x}$ and $g(x)={3}^{x}.$

விடை தெரியப்படுத்து

We will use point plotting to graph the functions.
Graph: $f(x)={4}^{x}.$

விடை தெரியப்படுத்து
Graph: $g(x)={5}^{x}.$

விடை தெரியப்படுத்து
On the same coordinate system, graph $f(x)={(\frac{1}{2})}^{x}$ and $g(x)={(\frac{1}{3})}^{x}.$

விடை தெரியப்படுத்து

We will use point plotting to graph the functions.
Graph: $f(x)={(\frac{1}{4})}^{x}.$

விடை தெரியப்படுத்து
Graph: $g(x)={(\frac{1}{5})}^{x}.$

விடை தெரியப்படுத்து
On the same coordinate system graph $f(x)={2}^{x}$ and $g(x)={2}^{x+1}.$

விடை தெரியப்படுத்து

We will use point plotting to graph the functions.
On the same coordinate system, graph: $f(x)={2}^{x}$ and $g(x)={2}^{x-1}.$

விடை தெரியப்படுத்து
On the same coordinate system, graph: $f(x)={3}^{x}$ and $g(x)={3}^{x+1}.$

விடை தெரியப்படுத்து
On the same coordinate system graph $f(x)={3}^{x}$ and $g(x)={3}^{x}-2.$

விடை தெரியப்படுத்து

We will use point plotting to graph the functions.
On the same coordinate system, graph: $f(x)={3}^{x}$ and $g(x)={3}^{x}+2.$

விடை தெரியப்படுத்து
On the same coordinate system, graph: $f(x)={4}^{x}$ and $g(x)={4}^{x}-2.$

விடை தெரியப்படுத்து
Solve: ${3}^{2x-5}=27.$
Solve: ${3}^{3x-2}=81.$

விடை தெரியப்படுத்து

$x=2$
Solve: ${7}^{x-3}=7.$

விடை தெரியப்படுத்து

$x=4$

Solve $\frac{{e}^{{x}^{2}}}{{e}^{3}}={e}^{2x}$.

விடை தெரியப்படுத்து

	$\ \frac{{e}^{{x}^{2}}}{{e}^{3}}={e}^{2x}$
Use the Property of Exponents: $\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}.$	$\ {e}^{{x}^{2}-3}={e}^{2x}$
Write a new equation by setting the exponents equal.	$\ {x}^{2}-3=2x\$
Solve the equation.	${x}^{2}-2x-3=0\$
	$(x-3)(x+1)=0\$
	$x=3,\ x=-1$
Check the solutions.

Solve: $\frac{{e}^{{x}^{2}}}{{e}^{x}}={e}^{2}.$

விடை தெரியப்படுத்து

$x=-1,x=2$
Solve: $\frac{{e}^{{x}^{2}}}{{e}^{x}}={e}^{6}.$

விடை தெரியப்படுத்து

$x=-2,x=3$

A total of $\text{\$}10,000$ was invested in a college fund for a new grandchild. If the interest rate is $5\text{\%},$ how much will be in the account in 18 years by each method of compounding?

ⓐ compound quarterly

ⓑ compound monthly

விடை தெரியப்படுத்து

	$\ A=?$
Identify the values of each variable in the formulas.	$\ P=\text{\$}10,000$
Remember to express the percent as a decimal.	$\ r=0.05$
	$\ t=18\ \text{years}$

ⓐ

For quarterly compounding, $n=4$. There are 4 quarters in a year.	$\ A=P{(1+\frac{r}{n})}^{nt}$
Substitute the values in the formula.	$\ A=10,000{(1+\frac{0.05}{4})}^{4\cdot 18}$
Compute the amount. Be careful to consider the order of operations as you enter the expression into your calculator.	$\ A=\text{\$}24,459.20$

ⓑ

For monthly compounding, $n=12$. There are 12 months in a year.	$\ A=P{(1+\frac{r}{n})}^{nt}$
Substitute the values in the formula.	$\ A=10,000{(1+\frac{0.05}{12})}^{12\cdot 18}$
Compute the amount.	$\ A=\text{\$}24,550.08$

ⓒ

For compounding continuously,	$\ A=P{e}^{rt}$
Substitute the values in the formula.	$\ A=10,000{e}^{0.05\cdot 18}$
Compute the amount.	$\ A=\text{\$}24,596.03$

Angela invested $\text{\$}15,000$ in a savings account. If the interest rate is $4\text{\%},$ how much will be in the account in 10 years by each method of compounding?
ⓐ compound quarterly
ⓑ compound monthly
ⓒ compound continuously

விடை தெரியப்படுத்து

ⓐ $\text{\$}22,332.96$
ⓑ $\text{\$}22,362.49$ ⓒ $\text{\$}22,377.37$
Allan invested $10,000 in a mutual fund. If the interest rate is $5\text{\%},$ how much will be in the account in 15 years by each method of compounding?
ⓐ compound quarterly
ⓑ compound monthly
ⓒ compound continuously

விடை தெரியப்படுத்து

ⓐ $21,071.81 ⓑ $21,137.04
ⓒ $21,170.00

Chris is a researcher at the Center for Disease Control and Prevention and he is trying to understand the behavior of a new and dangerous virus. He starts his experiment with 100 of the virus that grows continously at a rate of 25% per hour. He will check on the virus in 24 hours. How many viruses will he find?

விடை தெரியப்படுத்து

Identify the values of each variable in the formulas.	$\ A=?$
Be sure to put the percent in decimal form.	${A}_{0}=100$
Be sure the units match—the rate is per hour and the time is in hours.	$\ r=0.25\text{/hour}$
	$\ t=24\ \text{hours}$

Substitute the values in the formula: $A={A}_{0}{e}^{rt}$.	$A=100{e}^{0.25\cdot 24}$
Compute the amount.	$A=40,342.88$
Round to the nearest whole virus.	$A=40,343$
	The researcher will find 40,343 viruses.

Another researcher at the Center for Disease Control and Prevention, Lisa, is studying the growth of a bacteria. She starts her experiment with 50 of the bacteria that grows at a rate of $15\text{\%}$ per hour. She will check on the bacteria every 8 hours. How many bacteria will she find in 8 hours?

விடை தெரியப்படுத்து

She will find 166 bacteria.
Milan, a biologist is observing the growth pattern of a virus. They start with 100 of the virus that grows at a rate of $10\text{\%}$ per hour. They will check on the virus in 24 hours. How many viruses will they find?

விடை தெரியப்படுத்து

They will find 1,102 viruses.
$f(x)={2}^{x}$

விடை தெரியப்படுத்து
$g(x)={3}^{x}$
$f(x)={6}^{x}$

விடை தெரியப்படுத்து
$g(x)={7}^{x}$
$f(x)={(1.5)}^{x}$

விடை தெரியப்படுத்து
$g(x)={(2.5)}^{x}$
$f(x)={(\frac{1}{2})}^{x}$

விடை தெரியப்படுத்து
$g(x)={(\frac{1}{3})}^{x}$
$f(x)={(\frac{1}{6})}^{x}$

விடை தெரியப்படுத்து
$g(x)={(\frac{1}{7})}^{x}$
$f(x)={(0.4)}^{x}$

விடை தெரியப்படுத்து
$g(x)={(0.6)}^{x}$
$f(x)={4}^{x},\ g(x)={4}^{x-1}$

விடை தெரியப்படுத்து

Symbols used here

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

\infty

infinity
Not a number: "grows without bound" in limits and intervals.

\approx

approximately equal
Equal to the precision shown, not exactly.

\neq

not equal
The two sides are different.

\sup,\ \inf

supremum, infimum
Least upper bound, greatest lower bound.

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

|x|

absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.

imaginary unit
i² = −1.

\log_b x,\ \ln x

logarithm, natural log
The exponent b must be raised to for x; ln uses base e.

\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}

number sets
Naturals, integers, rationals, reals, complex numbers.

How to: Evaluate and Graph Exponential Functions

Graph exponential functions
Solve Exponential equations
Use exponential models in applications
Write both sides of the equation with the same base, if possible.
Write a new equation by setting the exponents equal.
Solve the equation.
Check the solution.
Write both sides of the equation with the same base, if possible.

Questions people ask

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

உங்களை முயற்சிக்கவும்

Parts of this page are adapted from OpenStax Intermediate Algebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

மேலும் Algebra

Linear equations Quadratic equations Systems of equations Inequalities Factoring Expanding Simplifying expressions Functions and graphs Exponential and logarithmic equations Polynomial equations Absolute value

Domain	\((\text{-}\infty ,\infty )\)
Range	\((0,\infty )\)
x-intercept	None
y-intercept	\((0,1)\)
Contains	\((1,a),\ (-1,\frac{1}{a})\)
Asymptote	\(x\)-axis, the line \(y=0\)

	\(\ \frac{{e}^{{x}^{2}}}{{e}^{3}}={e}^{2x}\)
Use the Property of Exponents: \(\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}.\)	\(\ {e}^{{x}^{2}-3}={e}^{2x}\)
Write a new equation by setting the exponents equal.	\(\ {x}^{2}-3=2x\\)
Solve the equation.	\({x}^{2}-2x-3=0\\)
	\((x-3)(x+1)=0\\)
	\(x=3,\ x=-1\)
Check the solutions.

	\(\ A=?\)
Identify the values of each variable in the formulas.	\(\ P=\text{\$}10,000\)
Remember to express the percent as a decimal.	\(\ r=0.05\)
	\(\ t=18\ \text{years}\)

For quarterly compounding, \(n=4\). There are 4 quarters in a year.	\(\ A=P{(1+\frac{r}{n})}^{nt}\)
Substitute the values in the formula.	\(\ A=10,000{(1+\frac{0.05}{4})}^{4\cdot 18}\)
Compute the amount. Be careful to consider the order of operations as you enter the expression into your calculator.	\(\ A=\text{\$}24,459.20\)

For monthly compounding, \(n=12\). There are 12 months in a year.	\(\ A=P{(1+\frac{r}{n})}^{nt}\)
Substitute the values in the formula.	\(\ A=10,000{(1+\frac{0.05}{12})}^{12\cdot 18}\)
Compute the amount.	\(\ A=\text{\$}24,550.08\)

when \(a>1\)	when \(0
Domain	\((\text{-}\infty ,\infty )\)	Domain	\((\text{-}\infty ,\infty )\)
Range	\((0,\infty )\)	Range	\((0,\infty )\)
\(x\)-intercept	none	\(x\)-intercept	none
\(y\)-intercept	\((0,1)\)	\(y\)-intercept	\((0,1)\)
Contains	\((1,a),\ (-1,\frac{1}{a})\)	Contains	\((1,a),\ (-1,\frac{1}{a})\)
Asymptote	\(x\)-axis, the line \(y=0\)	Asymptote	\(x\)-axis, the line \(y=0\)
Basic shape	increasing	Basic shape	decreasing

For compounding continuously,	\(\ A=P{e}^{rt}\)
Substitute the values in the formula.	\(\ A=10,000{e}^{0.05\cdot 18}\)
Compute the amount.	\(\ A=\text{\$}24,596.03\)