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5. Exponential Functions
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United States of AmericaFL
Grade 912

5.03 Exponential relationships

Lesson
Practice
Lesson

Concept summary

Exponential relationships include any relations where the outputs change by a constant factor for consistent changes in x, and form a pattern.

In the table, we can see the change in output is increasing by a factor of 3, and can describe this pattern as "the number triples each time".

x0123
y13927

This relationship can be shown on a coordinate plane, with the curve passing through the points from the table.

-4
-3
-2
-1
1
2
3
4
x
5
10
15
20
25
30
y
  • This graph shows an exponential relationship.
  • y approaches a minimum value as x decreases and approaches \infty as x increases.

An exponential relationship can be modeled by a function with a variable in the exponent, known as an exponential function:

\displaystyle f\left(x\right)=ab^x
\bm{a}
The initial value
\bm{b}
The growth or decay factor

The initial value is the output value when x=0, and the growth or decay factor is the constant factor.

Worked examples

Example 1

Consider the following pattern:

Three groups of connected squares are shown in the pattern. The group labeled Step 1 has 2 squares. The group labeled Step 2 has 4 squares. The group labeled Step 3 has 8 squares.
a

Describe the pattern in words.

Approach

We can see that the first step is made up of 2 squares, the second step is made up of 4 squares, and the third step 8 squares. If we only considered the first two steps we would not know if this relationship was linear or exponential, we would just know it has increased by 2. By considering the increase from step 2 to step 3, we can see that it has increased by 4 squares, so we know the relationship is not linear.

Solution

The number of squares doubles each step.

Reflection

We could have constructed a table of values showing the step number and the number of squares in each step to see the pattern in another form.

b

Determine the number of squares the next step if the pattern continues.

Approach

Using the pattern we described in part (a), we have to double the number of squares in step 3 to find the number of squares in the next step.

Solution

2\cdot 8 = 16

There are 16 squares in the next step.

Example 2

For the following exponential function:

x1234
f\left(x\right)525125625
a

Identify the growth factor.

Approach

We can find the growth factor by dividing a term by the previous term, that is by evaluating. We can see that when x=1, f\left(x\right)=5 and when x=2, f\left(x\right)=25.

Solution

\dfrac{25}{5}=5

Reflection

We could have chosen other values and arrived at the same result. For example \dfrac{125}{25}=5.

b

Determine the value of f\left(5\right).

Approach

Using the growth factor found in part (a), we know that as x increases by 1, f\left(x\right) increases by a factor of 5. This means f\left(5\right)=5\times f\left(4\right).

Solution

f\left(5\right)=5\times 128 = 3125

Example 3

A large puddle of water starts evaporating when the sun shines directly on it. The amount of water in the puddle over time is shown in the table.

Hours since sun came outVolume in mL
01024
1512
2256
3
464
5

Assuming the relationship is exponential, complete the table and describe the relationship between time and volume.

Approach

We can find the value of b by dividing the amount of water in the puddle after one hour by the amount that was present at the start. Using this value for b, we can then find the missing values.

Solution

b=\dfrac{512}{1024}=\dfrac{1}{2}

Using this value for b, we know that the volume after 3 hours will be half of 256 and the time after 5 hours will be half of 64.

Hours since sun came outVolume in mL
01024
1512
2256
3128
464
532

Reflection

This exponential relation is an example of one that decreases over time. We can see that it represents decay instead of growth because in this case b is less than one.

Outcomes

MA.912.F.1.1

Given an equation or graph that defines a function, determine the function type. Given an input-output table, determine a function type that could represent it.

MA.912.F.1.8

Determine whether a linear, quadratic or exponential function best models a given real-world situation.

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