Topics
7. Quadratic Functions
topic badge
United States of AmericaVirginia
Algebra 1

7.05 Compare linear, quadratic, and exponential functions

Lesson
Practice

Comparing functions

In this lesson, we will use our prior knowledge of linear, quadratic, and exponential functions to identify key features and compare various functions represented in different ways.

Exploration

Consider the table:

xy=3xy=3x^2y=3^x
1333
26129
392727
515125243
  1. Compare the three functions and how they change as x increases.

The way a function is represented can affect the characteristics we are able to identify for the function. Different representations can highlight or hide certain characteristics. Remember that key features of functions include:

  • domain and range
  • x- and y-intercepts
  • maximum or minimum value(s)
  • how the function increases or decreases
  • vertex

One way to compare functions is to look at growth rates as the x-values increase over regular intervals. In order to compare the growth rates of quadratics with those of exponential or linear functions, we will look only at the half of the quadratic that is increasing.

-2
-1
1
2
3
4
5
6
x
10
20
30
y

Notice starting at x=0, g(x) is greater than h(x) and is increasing at a greater rate. But, as x continues to increase, the quadratic function g(x) is increasing at a slower rate than the exponential function, and eventually the exponential function will overtake the quadratic function.

Notice that no matter what the intercepts are, an exponential growth function will always exceed a linear or quadratic growth function as values of x become larger.

Examples

Example 1

Which of the following functions increases the fastest for very large values of x?

  • y = 9 \cdot x
  • y = 3^x
  • y = 2x^2
  • y = 4^x
Worked Solution
Create a strategy

Remember that exponential functions (b>1) will always have a greater rate of change when compared to linear and quadratic functions as x increases toward infinity. Then, we must compare the constant factor b in the equation y = a \cdot b^x.

Apply the idea

As x increases toward infinity, we know that our greatest rate of change is from one of the exponential functions, y = 3^xor y = 4^x. Remembering our lesson on characteristics of exponential functions, a greater constant factor, b, will result in a greater rate of change as x continues to increase. Therefore, y = 4^x increases the fastest for very large values of x.

Reflect and check

How would you approach this problem if exponential equations had 2 different leading coefficients? For example, would y = 2 \cdot 4^x or y = 4 \cdot 2^x have a greater rate of change as x increases toward infinity?

Example 2

Consider the functions shown. Assume that the domain of f is all real numbers.

  • Function 1:

    x-1012345
    f\left(x\right)-3.75-2-0.251.53.2556.75

  • Function 2:

    -8
    -6
    -4
    -2
    2
    4
    6
    8
    x
    -4
    -2
    2
    4
    6
    8
    10
    12
    y
a

Determine which function has a higher y-intercept.

Worked Solution
Create a strategy

Remember that the y-intercept of a function occurs when x=0. We can use this to evaluate the y-intercept of f and identify the y-intercept of g.

Apply the idea

For f, we can see from the table that f\left(0\right) = -2.

For g, we can see from the graph that g\left(0\right) = -3.

So the y-intercept of f is the point \left(0,\,-2\right) and the y-intercept of g is the point \left(0,\,-3\right), and therefore f has a higher y-intercept.

b

Determine which function will be greater as x gets very large.

Worked Solution
Create a strategy

We can consider how quickly each function changes to get an idea of how it will behave for very large values of x.

Apply the idea

We can see that f\left(x\right) has a constant rate of change (slope) of 1.75. This is a linear function. From the graph, we can see that g\left(x\right) increases at an increasing rate as x increases. So, we can see that Function 2 will eventually surpass Function 1 as x gets very large.

Example 3

Consider functions representing three options to earn money one of the following ways:

A figure showing 3 options to earn money. Option 1 shows the statement: You are given 2 dollars each day. Option 2 shows a table with 2 columns titled Days and Total Amount and with 6 rows. The data is as follows: First column: 1, 2, 3, 4, 5,6; Second column: 1 dollar, 4 dollars, 9 dollars, 16 dollars, 25 dollars, 36 dollars. Option 3 shows a first quadrant coordinate plane with the x axis labeled Days and the y axis labeled Total Amount in dollars. The points (1, 2), (2, 4), (3, 8), (4, 16), and (5, 32) are plotted on the graph.

Note: Option 3 starts with \$2 on day one and doubles each day after this.

a

Find the equation that represents each option, where x is the number of days that have passed.

Worked Solution
Create a strategy

For each option, we can consider how the total amount of money changes as the days progress and derive an equation to represent the relationship.

Apply the idea

We can see that Option 1 has a constant rate of change regardless of the interval we considered. So, Option 1 can be represented by the linear function, f\left(x\right)=2x.

Now, observing the table of values for Option 2, we can see that the total amount is just the square of the number of days passed. So, Option 2 can be represented by the function f\left(x\right)=x^2.

Finally, the relationship for Option 3 is represented in the graph, but also described to us. Since we are told that the function starts at \$2 and is doubled each day, we can see that Option 3 is just represented by the function f\left(x\right)=2^x.

Reflect and check

If the relationship between the days passed and the total amount weren't directly obvious in Option 2, we could have tested the data provided in the table to rule out a linear or exponential relationship.

For a linear relationship, the rate of change between any two points must be equal. We can check that this wasn't true for Option 2. So, we could have then tested if it represented an exponential relationship.

For an exponential relationship, the ratio of between two points, a unit apart, must be equal. We can see that for Option 2, \dfrac{4}{1} \neq \dfrac{9}{4}.

Therefore, we could see that Option 2 represented neither a linear or exponential relationship.

b

Find the value of each option at 8 days, 12 days, and 14 days.

Worked Solution
Create a strategy

Construct a table of values with the amounts of money gained with each option.

Apply the idea
DaysOption 1 TotalOption 2 TotalOption 3 Total
1\$2\$1\$2
2\$4\$4\$4
3\$6\$9\$8
4\$8\$16\$16
5\$10\$25\$32
6\$12\$36\$64
7\$14\$49\$128
8\$16\$64\$256
9\$18\$81\$512
10\$20\$100\$1\,024
11\$22\$121\$2\,048
12\$24\$144\$4\, 096
13\$26\$169\$8\,192
14\$28\$196\$16 \,384

At 8 days, Option 1 will make \$16, Option 2 will make \$64, and Option 3 will make \$256.

At 12 days, Option 1 will make \$24, Option 2 will make \$144, and Option 3 will make \$4\,096.

At 14 days, Option 1 will make \$28, Option 2 will make \$196, and Option 3 will make \$16\,384.

Reflect and check

We could calculate the total amount of money on days 8,\,12 and 14 using the functions found in part (a), instead of constructing a table.

c

Determine which option will be greater for larger and larger values of x.

Worked Solution
Create a strategy

Use the table comparison from part (b) to determine which option will be greater for larger and larger values of x.

Apply the idea

As x gets larger and larger, we can see that Option 3, the exponential option, will be far greater than Options 1 or 2.

Reflect and check

An exponential function will always exceed a linear or quadratic function as values of x become larger.

Idea summary

It is important to be able to compare the key features of functions whether they are represented in similar or different ways:

  • domain and range
  • x- and y-intercepts
  • maximum or minimum value(s)
  • how the function increases or decreases
  • vertex

Outcomes

A.F.1

The student will investigate, analyze, and compare linear functions algebraically and graphically, and model linear relationships.

A.F.1a

Determine and identify the domain, range, zeros, slope, and intercepts of a linear function, presented algebraically or graphically, including the interpretation of these characteristics in contextual situations.

A.F.1f

Graph a linear function in two variables, with and without the use of technology, including those that can represent contextual situations.

A.F.1g

For any value, x, in the domain of f, determine f(x), and determine x given any value f(x) in the range of f, given an algebraic or graphical representation of a linear function.

A.F.2

The student will investigate, analyze, and compare characteristics of functions, including quadratic and exponential functions, and model quadratic and exponential relationships.

A.F.2b

Given an equation or graph, determine key characteristics of a quadratic function including x-intercepts (zeros), y-intercept, vertex (maximum or minimum), and domain and range (including when restricted by context); interpret key characteristics as related to contextual situations, where applicable.

A.F.2c

Graph a quadratic function, f(x), in two variables using a variety of strategies, including transformations f(x) + k and kf(x), where k is limited to rational values.

A.F.2e

Given an equation or graph of an exponential function in the form y = ab^(x) (where b is limited to a natural number), interpret key characteristics, including y-intercepts and domain and range; interpret key characteristics as related to contextual situations, where applicable.

A.F.2f

Graph an exponential function, f(x), in two variables using a variety of strategies, including transformations f(x) + k and kf(x), where k is limited to rational values.

A.F.2g

For any value, x, in the domain of f, determine f(x) of a quadratic or exponential function. Determine x given any value f(x) in the range of f of a quadratic function. Explain the meaning of x and f(x) in context.

A.F.2h

Compare and contrast the key characteristics of linear functions (f(x) = x), quadratic functions (f(x) = x^(2)), and exponential functions (f(x) = b^(x)) using tables and graphs.

What is Mathspace

About Mathspace