6.1 Exponential Functions

Evaluate the function $f \left(x\right) = 3^{x}$ for the given values

1. ⓐ $f \left(2\right)$
2. ⓑ $f \left(-1\right)$
3. ⓒ $f \left(2 h\right)$

Which of the following equations are not exponential functions?

1. $f \left(x\right) = 4^{3 \left(x - 2\right)}$
2. $g \left(x\right) = x^{3}$
3. $h \left(x\right) = \left(\frac{1}{3}\right)^{x}$
4. $j \left(x\right) = \left(- 2\right)^{x}$

Which of the following equations represent exponential functions?

• $f \left(x\right) = 2 x^{2} - 3 x + 1$
• $g \left(x\right) = 0.875^{x}$
• $h \left(x\right) = 1.75 x + 2$
• $j \left(x\right) = 1095.6^{- 2 x}$

Let $f \left(x\right) = 5 \left(3\right)^{x + 1} .$ Evaluate $f \left(2\right)$ without using a calculator.

Let $f \left(x\right) = 8 \left(1.2\right)^{x - 5} .$ Evaluate $f \left(3\right)$ using a calculator. Round to four decimal places.

At the beginning of this section, we learned that the population of India was about $1.25$ billion in the year 2013, with an annual growth rate of about $1.2 \% .$ This situation is represented by the growth function $P \left(t\right) = 1.25 \left(1.012\right)^{t} ,$ where $t$ is the number of years since $2013.$ To the nearest thousandth, what will the population of India be in $\text{2031}?$

The population of China was about 1.39 billion in the year 2013, with an annual growth rate of about $0.6 \% .$ This situation is represented by the growth function $P \left(t\right) = 1.39 \left(1.006\right)^{t} ,$ where $t$ is the number of years since $2013.$ To the nearest thousandth, what will the population of China be for the year 2031? How does this compare to the population prediction we made for India in the example?

A wolf population is growing exponentially. In 2011, $129$ wolves were counted. By $\text{2013},$ the population had reached 236 wolves. What two points can be used to derive an exponential equation modeling this situation? Write the equation representing the population $N$ of wolves over time $t .$

Given the two points $\left(1 , 3\right)$ and $\left(2 , 4.5\right) ,$ find the equation of the exponential function that passes through these two points.

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

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

If we invest $3,000 in an investment account paying 3% interest compounded quarterly, how much will the account be worth in 10 years?

An initial investment of $100,000 at 12% interest is compounded weekly (use 52 weeks in a year). What will the investment be worth in 30 years?

A 529 Plan is a college-savings plan that allows relatives to invest money to pay for a child’s future college tuition; the account grows tax-free. Lily wants to set up a 529 account for her new granddaughter and wants the account to grow to $40,000 over 18 years. She believes the account will earn 6% compounded semi-annually (twice a year). To the nearest dollar, how much will Lily need to invest…

Refer to the example. To the nearest dollar, how much would Lily need to invest if the account is compounded quarterly?

Calculate $e^{3.14} .$ Round to five decimal places.

Use a calculator to find $e^{- 0.5} .$ Round to five decimal places.

A person invested $1,000 in an account earning a nominal 10% per year compounded continuously. How much was in the account at the end of one year?

A person invests $100,000 at a nominal 12% interest per year compounded continuously. What will be the value of the investment in 30 years?

Radon-222 decays at a continuous rate of 17.3% per day. How much will 100 mg of Radon-222 decay to in 3 days?

Using the data in the example, how much radon-222 will remain after one year?

Explain why the values of an increasing exponential function will eventually overtake the values of an increasing linear function.

The Oxford Dictionary defines the word nominal as a value that is “stated or expressed but not necessarily corresponding exactly to the real value.”Oxford Dictionary. http://oxforddictionaries.com/us/definition/american_english/nomina. Develop a reasonable argument for why the term nominal rate is used to describe the annual percentage rate of an investment account that compounds interest.

Identify whether the statement represents an exponential function. Explain.

A population of bacteria decreases by a factor of $\frac{1}{8}$ every $24$ hours.

Identify whether the statement represents an exponential function. Explain.

For each training session, a personal trainer charges his clients $\$ 5$ less than the previous training session.

Consider this scenario: For each year $t ,$ the population of a forest of trees is represented by the function $A \left(t\right) = 115 \left(1.025\right)^{t} .$ In a neighboring forest, the population of the same type of tree is represented by the function $B \left(t\right) = 82 \left(1.029\right)^{t} .$ (Round answers to the nearest whole number.)

Which forest’s population is growing at a faster rate?

Consider this scenario: For each year $t ,$ the population of a forest of trees is represented by the function $A \left(t\right) = 115 \left(1.025\right)^{t} .$ In a neighboring forest, the population of the same type of tree is represented by the function $B \left(t\right) = 82 \left(1.029\right)^{t} .$ (Round answers to the nearest whole number.)

Assuming the population growth models continue to represent the growth of the forests, which forest will have a greater number of trees after $20$ year…

Consider this scenario: For each year $t ,$ the population of a forest of trees is represented by the function $A \left(t\right) = 115 \left(1.025\right)^{t} .$ In a neighboring forest, the population of the same type of tree is represented by the function $B \left(t\right) = 82 \left(1.029\right)^{t} .$ (Round answers to the nearest whole number.)

Discuss the above results from the previous four exercises. Assuming the population growth models continue to represent the growth of the forests, w…

Determine whether the equation represents exponential growth, exponential decay, or neither. Explain.

$y = 220 \left(1.06\right)^{x}$

Determine whether the equation represents exponential growth, exponential decay, or neither. Explain.

$y = 11 , 701 \left(0.97\right)^{t}$

Find the formula for an exponential function that passes through the two points given.

$\left(0 , 2000\right)$ and $\left(2 , 20\right)$

Find the formula for an exponential function that passes through the two points given.

$\left(- 2 , 6\right)$ and $\left(3 , 1\right)$

Determine whether the table could represent a function that is linear, exponential, or neither. If it appears to be exponential, find a function that passes through the points.

Determine whether the table could represent a function that is linear, exponential, or neither. If it appears to be exponential, find a function that passes through the points.

Determine whether the table could represent a function that is linear, exponential, or neither. If it appears to be exponential, find a function that passes through the points.

Use the compound interest formula, $A \left(t\right) = P \left(1 + \frac{r}{n}\right)^{n t} .$

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

Use the compound interest formula, $A \left(t\right) = P \left(1 + \frac{r}{n}\right)^{n t} .$

An account is opened with an initial deposit of $6,500 and earns $3.6 \%$ interest compounded semi-annually. What will the account be worth in $20$ years?

Use the compound interest formula, $A \left(t\right) = P \left(1 + \frac{r}{n}\right)^{n t} .$

Solve the compound interest formula for the principal, $P$ .

Use the compound interest formula, $A \left(t\right) = P \left(1 + \frac{r}{n}\right)^{n t} .$

How much more would the account in the previous two exercises be worth if it were earning interest for $5$ more years?

Use the compound interest formula, $A \left(t\right) = P \left(1 + \frac{r}{n}\right)^{n t} .$

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.

Determine whether the equation represents continuous growth, continuous decay, or neither. Explain.

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

Determine whether the equation represents continuous growth, continuous decay, or neither. Explain.

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

Determine whether the equation represents continuous growth, continuous decay, or neither. Explain.

How much less would the account from Exercise 42 be worth after $30$ years if it were compounded monthly instead?

Evaluate each function. Round answers to four decimal places, if necessary.

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

Evaluate each function. Round answers to four decimal places, if necessary.

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

Evaluate each function. Round answers to four decimal places, if necessary.

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

Use a graphing calculator to find the equation of an exponential function given the points on the curve.

$\left(0 , 3\right)$ and $\left(3 , 375\right)$

Use a graphing calculator to find the equation of an exponential function given the points on the curve.

$\left(20 , 29.495\right)$ and $\left(150 , 730.89\right)$

Use a graphing calculator to find the equation of an exponential function given the points on the curve.

$\left(11,310.035\right)$ and $\left(25,356365.2\right)$

Repeat the previous exercise to find the formula for the APY of an account that compounds daily. Use the results from this and the previous exercise to develop a function $I \left(n\right)$ for the APY of any account that compounds $n$ times per year.

In an exponential decay function, the base of the exponent is a value between 0 and 1. Thus, for some number $b > 1 ,$ the exponential decay function can be written as $f \left(x\right) = a \cdot \left(\frac{1}{b}\right)^{x} .$ Use this formula, along with the fact that $b = e^{n} ,$ to show that an exponential decay function takes the form $f \left(x\right) = a \left(e\right)^{- n x}$ for some positive number $n$ .

The fox population in a certain region has an annual growth rate of 9% per year. In the year 2012, there were 23,900 fox counted in the area. What is the fox population predicted to be in the year 2020?

In the year 1985, a house was valued at $110,000. By the year 2005, the value had appreciated to $145,000. What was the annual growth rate between 1985 and 2005? Assume that the value continued to grow by the same percentage. What was the value of the house in the year 2010?

Jaylen wants to save $54,000 for a down payment on a home. How much will he need to invest in an account with 8.2% APR, compounding daily, in order to reach his goal in 5 years?

Alyssa opened a retirement account with 7.25% APR in the year 2000. Her initial deposit was $13,500. How much will the account be worth in 2025 if interest compounds monthly? How much more would she make if interest compounded continuously?