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Writing Equations of Exponential Functions

Author: Sophia

what's covered

In this lesson, you will write equations of exponential functions using information, which could include a verbal description or two points that are known. Specifically, this lesson will cover:

Writing Equations Using Percent Increase and Decrease
Writing Equations Given Two Solution Points

1. Writing Equations Using Percent Increase and Decrease

The goal of this challenge will be to write exponential functions in the form f open parentheses x close parentheses equals a times b to the power of x comma where is a nonzero number and b is a positive number not equal to 1.

Earlier, you may recall, we saw situations in which a constant percent increase or decrease can be modeled by an exponential function.

EXAMPLE

Suppose the value of a house is predicted to increase by 3.4% per year for the foreseeable future. If its value now is $280,000, then what is an exponential model for the value of the home x years from now?

The decimal version of the percent increase is 0.034. This means that the factor by which the value of the home increases from year to year is 1 plus 0.034 equals 1.034.

To calculate the value of the home after x years, start at 280000, then multiply 1.034 for each year after the starting point.

Since the starting value of the home is 280000, the function that describes the value of the home after x years is f open parentheses x close parentheses equals 280000 open parentheses 1.034 close parentheses to the power of x.

f open parentheses x close parentheses equals 280000 open parentheses 1.034 close parentheses to the power of x.

Here is another example where a quantity is decreasing over time.

EXAMPLE

After taking 100 milligrams of medicine, the quantity of medicine in one’s bloodstream decreases by 30% every hour. What is the exponential function that models the amount of medicine remaining in the bloodstream after t hours?

The decimal version of the percent decrease is 0.30. This means that the factor by which the amount of medicine in the bloodstream changes from one hour to the next is 1 minus 0.30 equals 0.70.

To calculate the amount of medicine in the bloodstream after t hours, start at 100 mg, then multiply 0.70 for each hour after the starting point.

Since there is initially 100 milligrams of medicine in the bloodstream, the function that describes the amount of medicine in the bloodstream after t hours is f open parentheses t close parentheses equals 100 open parentheses 0.70 close parentheses to the power of t.

f open parentheses t close parentheses equals 100 open parentheses 0.70 close parentheses to the power of t.

big idea

Given an exponential model of the form f open parentheses x close parentheses equals a times b to the power of x comma

considering for now a greater than 0 comma

we observe these behaviors:

When the value of increases as x increases.
When the value of decreases as x increases.

When we talk about graphs of exponential functions later in this challenge, we will discuss the behavior of f open parentheses x close parentheses

when

try it

The starting salary for an engineer at a large company is $70,000 for one year, with a 5.2% raise in salary per year.

Write an exponential function for the annual salary t years after starting.

f open parentheses t close parentheses equals 70000 open parentheses 1.052 close parentheses to the power of t

2. Writing Equations Given Two Solution Points

When the growth (or decay) rate is unknown, we need to find and b in the equation f open parentheses x close parentheses equals a times b to the power of x some other way. Since there are two values to find, we can use two solution points to write the equation.

EXAMPLE

Write the equation of an exponential function f open parentheses x close parentheses

that contains solution points open parentheses 0 comma space 12 close parentheses

and

open parentheses 2 comma space 3 close parentheses.

Start with f open parentheses x close parentheses equals a times b to the power of x.

Since

is a solution point, this means f open parentheses 0 close parentheses equals 12.

		Replace x with 0.
		Replace with 12, and simplify
		Solve for

Since

is a solution point, this means f open parentheses 2 close parentheses equals 3.

		Replace x with 2.
		Replace with 3 and with 12.
		Divide both sides by 12 and simplify.
		Apply the square root principle.
		The value of b must be positive according to the definition of an exponential function. Therefore, is not considered.

Now, the equation of the exponential function is f open parentheses x close parentheses equals 12 open parentheses 1 half close parentheses to the power of x.

f open parentheses x close parentheses equals 12 open parentheses 1 half close parentheses to the power of x.

try it

An exponential function f open parentheses t close parentheses

has solution points open parentheses 0 comma space 4 close parentheses

and

open parentheses 3 comma space 13.5 close parentheses.

Write the equation for f (t ).

f open parentheses t close parentheses equals 4 open parentheses 1.5 close parentheses to the power of t

Notice how the previous problems were very nice to solve since the point open parentheses 0 comma space f open parentheses 0 close parentheses close parentheses comma the y-intercept, was known. When the y-intercept is not known, more advanced algebraic techniques are required.

EXAMPLE

Determine the equation of an exponential function that contains solution points open parentheses short dash 1 comma space 27 close parentheses

open parentheses short dash 1 comma space 27 close parentheses

and

open parentheses 2 comma space 64 close parentheses.

Using the equation f open parentheses x close parentheses equals a times b to the power of x comma

substitute each solution pair in order to find

and b.

Since open parentheses short dash 1 comma space 27 close parentheses

is a solution point, it follows that f open parentheses short dash 1 close parentheses equals 27.

		Replace x with -1.
		Replace with 27, and rewrite as
		Rewrite the right-hand side as one fraction.

Since

is a solution point, it follows that f open parentheses 2 close parentheses equals 64.

		Replace x with 2.
		Replace with 64.

At this point, we have two equations in the unknowns

and b. These are not linear equations as we are accustomed to, but we can still solve using the substitution method.

Consider the equation 27 equals a over b.

Multiplying both sides by b gives a equals 27 b.

This can now be substituted into the other equation, which can then be solved for b.

		Start with this equation.
		Substitute
		Simplify the right-hand side.
		Divide both sides by 27.
		Take the cube root of both sides.
		Simplify.

Thus,

Substituting into the equation a equals 27 b comma

we find

a equals 27 open parentheses 4 over 3 close parentheses equals 36.

Therefore, the equation of the exponential function is f open parentheses x close parentheses equals 36 open parentheses 4 over 3 close parentheses to the power of x.

f open parentheses x close parentheses equals 36 open parentheses 4 over 3 close parentheses to the power of x.

watch

Check out this video that demonstrates how to write the equation of the exponential function that passes through open parentheses short dash 2 comma space 25 over 48 close parentheses

open parentheses short dash 2 comma space 25 over 48 close parentheses

and

open parentheses 2 comma space 27 over 25 close parentheses.

Video Transcription

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Welcome. I'm glad you joined us. We are going to take and use an example to show how to write the equation of an exponential function that contains solution points that are given to us. And we want to write the answer in f of x is equal to a times b to the x form.

To do this, what we are going to do is first take our function f of x equal a times b to the x form and one of our ordered pairs to start with, and make some substitutions to generate an equation with just a and b.

So first of all, what we want to do is look at what happens if we take out the x and put negative 2 in its place in this form. Well, that would give me a times b to the negative 2 power.

Now let's look at what it means for the ordered pair. Well, the ordered pair is telling me that if I have our function, and I put negative 2 in for x, I will get out a 25 over 48 for the output. Well, that means, then, that these two quantities are equal. So I have a times b to the negative 2 is equal to 25 over 48.

Now we are going to go to the other ordered pair and do the same process. So we start with our f of x is equal to a times b to the x. f of 2 is a times-- we'll take out the x and put in 2, so a times b squared. But from the ordered pair, I have that a of 2 is equal to 27 over 25, the output of that ordered pair.

So then I get the equation that a times b squared is equal to 27 over 25. So now I have these two equations that I want to solve. a times b to the negative 2 is 25 over 48, and a times b squared equals 27 over 25.

Now these equations are not linear. But I can still solve them by substitution. Let's take this first equation and solve it for a. So remember, b to the negative 2-- well, a negative exponent is a reciprocal. So that's a times 1 over b squared is equal to 25 over 48.

Multiplying both sides by b squared, I get a is equal to 25 over 48 times B. Squared. And that 25 over 48 times b squared, we are going to replace in the a of this other equation. So that gives us 25 over 48b squared, and then times the b squared. That is the other factor in the equation we're plugging it into. So that's the a that we plugged in, and then times the other factor that's there. And that equals 27 over 25.

Well, this gives me 25 over 48 times b to the fourth is equal to 27 over 25. Solve for b to the fourth by multiplying both sides by 48 over 25, and I get b to the fourth is equal to 1,296 over 625.

Now we want to take the fourth root of both sides to solve for b. And remember, when we do that, we have to remember the plus or minus. So our b is equal to plus or minus 6/5.

However, if you recall, b is the base of our exponential part of the function, and the base of an exponential function has to be positive. So for this, we are only going to consider the positive. So we'll just go ahead and erase the minus off of that.

So that's our b. b is equal to 6/5.

Next up, we still need to find a. So we can go to any of these equations that we have for what a is equal to and solve for the value for a. We are going to look at the part that's highlighted in green. So we have a is equal to 25 over 48. And notice, that's times b squared. So we're going to take our 6/5 squared.

And this gives me a is equal to 25 over 48 times 36 over 25. And that simplifies to the value of 3/4.

So now we have our a and we have our b. Our function f of x is equal to a, which is the 3/4, times b, which is equal to the 6/5, to the x power. And that's how you write the equation of the exponential function when you're given two points that it contains.

try it

Consider an exponential function that contains solution points open parentheses short dash 2 comma space 27 close parentheses

and

open parentheses 2 comma space 1 third close parentheses.

Write the equation of the exponential function.

f open parentheses x close parentheses equals 3 open parentheses 1 third close parentheses to the power of x

summary

In this lesson, you learned how to write the equation of an exponential function in the form f open parentheses x close parentheses equals a times b to the power of x comma

where

is a nonzero number and b is a positive number not equal to 1, using two different methods: If known, you can use the starting value and corresponding percent increase and decrease, or you can write the equation given two solution points. It is important to note that when solving for the base b, remember that b has to be positive.

SOURCE: THIS WORK IS ADAPTED FROM PRECALCULUS BY JAY ABRAMSON. ACCESS FOR FREE AT OPENSTAX.ORG/BOOKS/PRECALCULUS/PAGES/1-INTRODUCTION-TO-FUNCTIONS