Topics
5. Linear and Exponential Models
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United States of America
Grades 9-12

5.06 Comparing linear and exponential functions

Lesson
Practice

Introduction

We explored the key features of linear functions in lesson  3.05 Graphing linear functions  and the key features of exponential functions in lesson  5.01 Exponential functions  . Both function types have similar characteristics, but we will explore their differences in this lesson.

Comparing linear and exponential functions

Key features of a function are useful in helping to sketch the function, as well as to interpret information about the function in a given context.

The characteristics, or key features, of a function include its:

  • domain and range

  • x- and y-intercepts

  • maximum or minimum value(s)

  • rate of change over specific intervals

  • end behavior

  • positive and negative intervals

  • increasing and decreasing intervals

Exploration

Leilani and Koda each open a bank account with \$100. Leilani's account will earn 3\% interest every month. Koda's account will earn \$9 every month.

  1. Who will have more money in the short term?
  2. Who will have more money in the long term?
  3. How do the rates of change differ?

We can use key features to compare linear and exponential functions. Many of their features are similar, but their rates of change are different. A linear function has a constant rate of change while an exponential function has a constant percent rate of change.

This means that we are adding the same number to each output of a linear function, but we are multiplying the same number to each output of an exponential function. Multiplication grows faster than addition, so a quantity increasing exponentially will always exceed a quantity increasing linearly over time.

Examples

Example 1

Consider the two functions shown in the graphs below.

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a

State the intercepts of each function.

Worked Solution
Create a strategy

We need to look for both the x-intercepts and the y-intercepts. We also need to state them as ordered pairs.

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Apply the idea

Both functions have a y-intercept at \left(0, -1\right).

Also, both functions have an x-intercept at \left(1, 0\right).

b

Compare the end behavior of the two functions.

Worked Solution
Apply the idea

On the right side, both functions take larger and larger positive values as x gets further from zero. That is, as x \to \infty, y \to \infty for both functions.

A line passing through points (1,0) and (0, -1) plotted on a four quadrant coordinate plane. On the first quadrant, there is a vertically upward arrow labeled y approaches infinity and a right arrow labeled x approaches infinity. On the third quadrant, there is a vertically downward arrow labeled y approaches negative infinity and a left arrow labeled x approaches negative infinity.

On the left side, f\left(x\right) takes larger and larger negative values as x gets further from zero. That is, as x \to -\infty, y \to -\infty for f\left(x\right).

An exponential curve passing through points (1,0) and (0, -1) and has an asymptote at y = negative 2 plotted on a four quadrant coordinate plane. On the first quadrant, there is a vertically upward arrow labeled y approaches infinity and a right arrow labeled x approaches infinity. Just above the negative x axis, there is a left arrow labeled x approaches negative infinity. On the third quadrant, there is a left arrow labeled y approaches negative 2.

On the other hand, the values of g\left(x\right) get closer and closer to y=-2 as x gets further from zero on the left side. That is, as x \to -\infty, y \to -2 for g\left(x\right).

Reflect and check

Note that although both functions tend towards infinity to the right, the way they do so is different. The first function increases at a constant rate, while the second function increases at an increasing rate.

c

Using the graph of each function, find where f\left(x\right)=g\left(x\right).

Worked Solution
Create a strategy

We can look for the points where f\left(x\right) and g\left(x\right) would intersect if drawn on the same plane.

Apply the idea

If we draw f\left(x\right) on the same plane as g\left(x\right), we get:

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We can see that our two functions intersect at \left(0,-1\right) and \left(1,0\right). This means that at x=0 and x=1 the two functions are equal to eachother.

Reflect and check

If we tried to solve this equation algebraically, we would have to solve something of the form ab^{x}=mx+b. Since this is hard to solve, we can plot the graphs of each side of the equation and use the point of intersection to solve the equations instead.

d

Compare the average rate of change of each function over the following intervals:

  • -1<x<0
  • 0<x<1
  • 1<x<2
Worked Solution
Create a strategy

We calculate the average rate of change with the formula \dfrac{f\left(b\right)-f\left(a\right)}{b-a}.

Apply the idea
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For the linear function:

\dfrac{f(0)-f(1)}{0-(-1)}=\dfrac{-1-(-2)}{1}=1

\dfrac{f(1)-f(0)}{1-0}=\dfrac{0-(-1)}{1}=1

\dfrac{f(2)-f(1)}{2-1}=\dfrac{1-0}{1}=1

The average rate of change is constant. The values are increasing by 1 each time.

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For the exponential function:

\dfrac{f(0)-f(-1)}{0-(-1)}=\dfrac{-1-(-1.5)}{1}=0.5

\dfrac{f(1)-f(0)}{1-0}=\dfrac{0-(-1)}{1}=1

\dfrac{f(2)-f(1)}{2-1}=\dfrac{2-0}{1}=2

The average rate of change is not constant. The values are changing by a greater amount each time.

Although both functions are increasing, the exponential function is increasing at a faster rate.

Reflect and check

Over the interval before the first point of intersection, -1<x<0, the exponential function actually has an average rate of change that is less than the linear function. However, as x increased we saw a rapid increase in the average rate of change. So, the exponential function is still increasing at a faster rate.

Example 2

Consider the functions shown in the graph and table below.

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g\left(x\right)0.40960.5120.640.811.251.56251.95312.4414
a

State whether each function is linear or exponential.

Worked Solution
Create a strategy

We can see from the graph of f\left(x\right) that it forms a straight line.

For the table of g\left(x\right), we need to look at the differences or ratios of the outputs to determine if it in linear or exponential.

Apply the idea

We will start by finding the differences in the outputs of g\left(x\right).

0.512-0.4096=0.1024

0.64-0.512=0.128

0.8-0.64=0.16

Since there isn't a common difference, we know the function is not linear. Next, we will find the ratios of the outputs.

0.512\div 0.4096=1.25

0.64\div 0.512=1.25

0.8\div 0.64=1.25

Since the outputs have a common ratio, g\left(x\right) is exponential. Since f\left(x\right) increases by a constant amount, it is linear.

b

Compare the intervals where the function is increasing and decreasing for each function.

Worked Solution
Create a strategy

Since f\left(x\right) is linear, we know it will only increase or decrease over its domain. We can determine whether f\left(x\right) is increasing or decreasing by looking at its graph.

Similarly, exponential functions only increase or decrease over their domain. We can determine whether g\left(x\right) is increasing or decreasing by looking at the outputs in the table.

Apply the idea

Looking at the end behavior of the graph of f\left(x\right), we see that the y-values are growing as x increases.

Looking at the function values (outputs) of the table of g\left(x\right), the values are getting larger as x increases.

Both functions are increasing over their domain.

c

Determine which function will have a higher value as x increases.

Worked Solution
Create a strategy

We already know that f\left(x\right) increases linearly while g\left(x\right) increases exponentially. We can use what we know about the rates of change of linear and exponential functions to determine which one will exceed the other.

Apply the idea

Since f\left(x\right) will grow by a constant amount but g\left(x\right) will grow by a constant percentage, g\left(x\right) will have a higher value as x increases.

Reflect and check

Using technology to graph both functions and extending the x-axis, we can confirm that the exponential function will exceed the linear function for higher values of x.

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Example 3

Two objects are depreciating in value as shown in the table below:

Number of years01234
Object A\$7\,500\$6\,000\$4\,800\$3\,840\$3\,072
Object B\$12\,000\$9\,000\$6\,750\$5\,062.50\$3\,796.88
a

Determine whether each object is decreasing linearly or exponentially.

Worked Solution
Create a strategy

A linear function has a constant rate of change. An exponential function has a constant percent rate of change. We need to determine if the values of each object have a common difference or a common ratio.

Apply the idea

By observing the first 3 values of each object, we can see that the objects are decreasing by different amounts.

For object A:

6000-7500=-1500

4800-6000=-1200

For object B:

9000-12000=-3000

6750-9000=-2250

This means that neither function is linear, so let's see if they are exponential by finding the ratios of the amounts.

For object A:

6\,000\div 7\,500=0.8

4\,800\div 6\,000=0.8

3\,840\div 4\,800=0.8

The values do have a common ratio, so object A depreciates exponentially. For object B:

9\,000\div 12,000=0.75

6\,750\div 9\,000=0.75

5\,062.5\div 6\,750 =0.75

The outputs share a common ratio, so both functions are exponential.

Reflect and check

Remember that exponential functions are in the form y=ab^x. When 0<b<1,the function decays exponentially.

Exponential decay function for object A: y=7500\left(0.8\right)^x

Exponential decay function for object B: y=12000\left(0.75\right)^x

b

Describe the rate of change of each object.

Worked Solution
Create a strategy

We already know the constant factors of the exponential functions that model the decay, but let's describe the rate of change as a constant percentage. Remember: {\text{Decay rate}=1-\text{Decay factor}}.

Apply the idea

The decay factor of object A is 0.8, so the decay rate is r=1-0.8=0.2. Multiplying that by 100 gives us 20 \%.

The decay factor of object B is 0.75, so the decay rate is r=1-0.75=0.25. Multiplying that by 100 gives us 25 \%.

Object A depreciates by 20\% while object B depreciates by 25\%.

c

Determine which object will have a higher value after 10 years.

Worked Solution
Create a strategy

Both functions are decreasing at a decreasing rate. However, the function with the lower percentage will decrease slower than the other function over time.

Apply the idea

Since object A only loses 20\% of its value each year while object B loses 25\% of its value each year, Object A will decrease at a slower rate. This means Object A have a higher value after 10 years.

Reflect and check

We can verify our answer by using the functions we found in the reflection from part (a):

Object A: y=7500\left(0.8\right)^{10}\approx \$805.31

Object B: y=12000\left(0.75\right)^{10}\approx \$675.76

Idea summary

A linear function has a constant rate of change while an exponential function has a constant percent rate of change. A quantity increasing exponentially will always exceed a quantity increasing linearly over time.

Outcomes

A.REI.D.11

Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately.

F.IF.B.4

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

F.IF.B.6

Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

F.IF.C.9

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

F.LE.A.1.A

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

F.LE.A.3

Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

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