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6. Exponential Functions
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United States of AmericaTennessee
Integrated Math 1

6.01 Exponential patterns and sequences

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.

Exponential relationships are related to geometric sequences where the term number is the input, and can only be a positive integer, and the term value is the output.

Geometric sequence

A sequence of numbers in which each consecutive pair of numbers has a common ratio.

Example:

1, \, 3, \, 9, \,27, \ldots

For the geometric sequence, 1, \, 3, \, 9, \,27, \ldots, we "multiply each term by 3 to get the next term" or "triple the number each time". This is called the recursive process.

We can show this geometric sequence in a table of values. From this table we can see that if we divide one term value by the previous one, we will always get 3, the constant factor or common ratio.

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 constant factor

The initial value is the output value when x=0 and the constant factor can tell us about how quickly the output values are growing or shrinking.

For a geometric sequence, instead of listing out the terms, we can use an explicit equation related to the exponential function:

\displaystyle a\left(n\right)=ab^n
\bm{a}
The value of term 0
\bm{b}
The common ratio
\bm{n}
The term number

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

Write the geometric sequence for the number of squares.

Approach

A geometric sequence will be a list of terms where to get from one term to the next we multiply by the same number.

Solution

Geometric sequence: 2, \, 4, \, 8, \ldots

Reflection

If we know there are no more terms after the given ones, we don't need the \ldots at the end, but if the pattern could continue, we can show that with \ldots at the end.

b

Describe the recursive process 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. When describing the pattern it must work to go both from Step 1 to Step 2 and from Step 2 to Step 3.

Solution

The number of squares doubles each step.

Reflection

It is important to look at every step and not just the first two. If we only looked at the first two we might have said "add 2 each time" which would be incorrect as this would not work to get from Step 2 to Step 3. Since we were told this is a geometric sequence, it must be an exponential, not linear relationship.

c

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

Approach

Using the pattern we described in part (b), 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 constant factor.

Approach

We can find the constant factor by dividing a term by the previous term.

Solution

Since we already know this is an exponential function, we can use any pair of values whose x-values are 1 unit apart. For example: \dfrac{25}{5}=5

Reflection

We could have chosen other values and arrived at the same result. For example \dfrac{125}{25}=5 and \dfrac{625}{125}=5.

b

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

Approach

Using the constant 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\cdot f\left(4\right).

Solution

\displaystyle f\left(5\right)\displaystyle =\displaystyle 5\cdot f\left(4\right)Use the constant factor
\displaystyle =\displaystyle 5\cdot 128Substution property
\displaystyle =\displaystyle 3125Evaluate

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
a

Given that the relationship is exponential, complete the table of values.

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

Constant factor: 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
3\frac{256}{2}=128
464
5\frac{64}{2}=32

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.

b

Describe the relationship between time and volume.

Approach

We found that b=\dfrac{1}{2} previously, this tells us how we move from one term to the next.

Solution

The volume of water is halved every hour.

Reflection

We can also word this as, "For every increase in time by one hour, the amount of water is divided by two."

Outcomes

M1.F.BF.A.1

Build a function that describes a relationship between two quantities.*

M1.F.BF.A.1.A

Determine steps for calculation, a recursive process, or an explicit expression from a context.

M1.F.LE.A.1.A

Know that linear functions grow by equal differences over equal intervals and that exponential functions grow by equal factors over equal intervals.

M1.F.LE.A.1.C

Recognize situations in which a quantity grows or decays by a constant factor per unit interval relative to another.

M1.MP1

Make sense of problems and persevere in solving them.

M1.MP2

Reason abstractly and quantitatively.

M1.MP3

Construct viable arguments and critique the reasoning of others.

M1.MP4

Model with mathematics.

M1.MP6

Attend to precision.

M1.MP7

Look for and make use of structure.

M1.MP8

Look for and express regularity in repeated reasoning.

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