Topics
4. Modeling with Functions
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United States of America
Grades 9-12

4.02 Function families

Lesson
Practice

Introduction

Throughout Math 1, we learned about several types of functions and analyzed and compared their key features to one another. We will review those types of functions and their key features in this lesson. Grouping each function into its own family will help us as we learn several new types of functions throughout Math 2.

Function families

There are many types of functions, and we can group them into categories called function families. Some of the function families we explored in Math 1 are listed below:

Constant function

A function that has a constant rate of change of 0. A constant function can be written in the form f\left(x\right) = c where c is any real number.

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  • Domain: all real numbers

  • Range: \left\{c\right\}

  • Always a horizontal line

  • No x-intercepts

  • One y-intercept

  • Rate of change is always 0

Linear function

A function that has a constant rate of change. A linear function can be written in the form f\left(x\right) = mx + b

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  • Domain: all real numbers

  • Range: all real numbers

  • Always increasing or always decreasing

  • One x-intercepts

  • One y-intercept

  • Rate of change is always constant

Absolute value function

A function that contains a variable expression inside absolute value bars; a function of the form f\left(x\right) = a\left|x - h\right| + k

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  • Domain: all real numbers

  • Has an absolute maximum or minimum

  • Has a vertical line of symmetry separating the increasing and decreasing intervals

  • 0, 1, or 2 x-intercepts

  • One y-intercept

  • Rate of change is constant; negative for half of the function and positive for the other half

Quadratic function

A polynomial function of degree 2. A quadratic function can be written in the form f\left(x\right) = ax^2 + bx + c where a \neq 0

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  • Domain: all real numbers

  • Has an absolute maximum or minimum

  • Has a vertical line of symmetry separating the increasing and decreasing intervals

  • 0, 1, or 2 x-intercepts

  • One y-intercept

  • Rate of change is variable

Exponential function

A function that has a constant percent rate of change. An exponential function can be written in the form f\left(x\right) = ab^x where a \neq 0 and b>0

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  • Domain: all real numbers

  • Has a horizontal asymptote

  • Always increasing or always decreasing

  • 0 or 1 x-intercepts

  • One y-intercept

  • Rate of change is a constant percent

A key feature of a function is the way in which it increases and decreases, known as its rate of change. To get an idea of how the graph of a function changes, we can take the average rate of change over a specific interval of the domain.

To find the average rate of change from a given function over the interval a \leq x \leq b, we can find the change in the value of the dependent variable f\left(b\right)-f\left(a\right) per change in value in the independent variable b-a.

\text{Average rate of change}=\dfrac{f\left(b\right)-f\left(a\right)}{b-a}

Examples

Example 1

Determine the type of function represented by the following tables.

a
x012345
f\left(x\right)-7-3151-3
Worked Solution
Create a strategy

Since the x-values in the table are increasing by 1 each time, we can determine the type of function by observing the rate of change of the outputs of the function.

Apply the idea

We can determine the rate of change by finding the differences between the outputs:\begin{aligned}-3-\left(-7\right)&=4\\1-\left(-3\right)&=4\\5-1&=4\\1-5&=-4\\-3-1&=-4 \end{aligned}

Since part of the function has a positive constant rate of change and the other part is a negative constant rate of change, these values represent an absolute value function.

Reflect and check

Plotting the points on a graph can also help us easily identify that the points represent an absolute value function.

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Worked Solution
Create a strategy

Observing the outputs of the function, we can see that they are growing at an increasing rate. \begin{aligned}12-8&=4\\18-12&=6\\27-18&=9\\40.5-27&=13.5\end{aligned}

There are only 2 types of functions we know of with a variable rate of change: a quadratic function and an exponential function. A quadratic function will have both an increasing and decreasing interval, but this table may only show an increasing interval portion of a quadratic function. So, we need to take a closer look at the rate of change and see if it fits an exponential pattern.

Apply the idea

An exponential function has a constant percent rate of change. Since the x-values increase by 1 each time, we can find this rate of change by dividing the outputs.\begin{aligned} 12\div8&=1.5\\18\div 12&=1.5\\27\div 18&=1.5\\40.5\div 27&=1.5 \end{aligned}Since the ratio of the outputs is the same, this function is exponential.

Reflect and check

Recall from Math 1 \text{Common ratio}=1+\text{growth rate} which gives us a decimal for the growth rate. Then, we multiply the rate by 100 to find the constant percent of growth.\begin{aligned}1.5&=1+r\\0.5&=r\\r&=50\%\end{aligned}

This means as x increases by 1, f\left(x\right) increases by 50\%.

Example 2

Determine the types of functions that are in this piecewise function.

f(x) = \begin{cases} x+4, & x \lt 0 \\ -2, & 0 \leq x \lt 4 \\ 12-x^2, & x\geq 4 \end{cases}

Worked Solution
Create a strategy

There are 3 types of functions that make up this piecewise function:

  • y=x+4

  • y=-2

  • y=12-x^2

We can use the structure of each equation to determine the function family it belongs to.

Apply the idea

The first equation is of the first degree and in the form f\left(x\right)=mx+b. In this case, m=1 and b=4. Therefore, the function that defines the interval x<0 is linear.

The second equation is in the form f\left(x\right)=c. In this case, c=-2. Therefore, the function that defines the interval 0\leq x <4 is constant.

The third equation is of the second degree and in the form f\left(x\right)=ax^2+bx+c. In this case, a=-1, b=0, and c=12. Therefore, the function that defines the interval x\geq 4 is quadratic.

Reflect and check
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If we had looked at the graph alone, we may have mistakenly classified the functions. Note that the linear function could have been part of an absolute value function, and the piece of the quadratic that is shown also appears linear.

This is why it is important to consider multiple features of the function instead of relying on a graph alone.

Example 3

The graph shows the height of a baseball in feet after it is thrown.

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a

Find the average rate of change of the height of the ball in the following intervals:

  • 0\leq t\leq 1

  • 1\leq t\leq 2

  • 2\leq t\leq 3

  • 3\leq t\leq 4

Worked Solution
Create a strategy

The average rate of change is found by \dfrac{f\left(b\right)-f\left(a\right)}{b-a} where a is one endpoint of the interval and b is the other endpoint of the interval.

Apply the idea

  • 0\leq t\leq 1

    \displaystyle \dfrac{f\left(b\right)-f\left(a\right)}{b-a}\displaystyle =\displaystyle \dfrac{f\left(1\right)-f\left(0\right)}{1-0}Substitute a=0 and b=1
    \displaystyle =\displaystyle \dfrac{9-0}{1-0}Substitute f\left(0\right)=0 and f\left(1\right)=9
    \displaystyle =\displaystyle 9Evaluate the subtraction and division

    Between the time the ball is thrown and 1 second after it is thrown, the ball is traveling an average of 9\text{ ft/s}.

  • 1\leq t\leq 2

    \displaystyle \dfrac{f\left(b\right)-f\left(a\right)}{b-a}\displaystyle =\displaystyle \dfrac{f\left(2\right)-f\left(1\right)}{2-1}Substitute a=1 and b=2
    \displaystyle =\displaystyle \dfrac{12-9}{2-1}Substitute f\left(1\right)=9 and f\left(2\right)=12
    \displaystyle =\displaystyle 3Evaluate the subtraction and division

    Between 1 and 2 seconds after the ball is thrown, it is traveling an average of 3\text{ ft/s}.

  • 2\leq t\leq 3

    \displaystyle \dfrac{f\left(b\right)-f\left(a\right)}{b-a}\displaystyle =\displaystyle \dfrac{f\left(3\right)-f\left(2\right)}{3-2}Substitute a=2 and b=3
    \displaystyle =\displaystyle \dfrac{9-12}{3-2}Substitute f\left(2\right)=12 and f\left(3\right)=9
    \displaystyle =\displaystyle -3Evaluate the subtraction and division

    Between 2 and 3 seconds after the ball is thrown, it is traveling an average of -3\text{ ft/s}.

  • 3\leq t\leq 4

    \displaystyle \dfrac{f\left(b\right)-f\left(a\right)}{b-a}\displaystyle =\displaystyle \dfrac{f\left(4\right)-f\left(3\right)}{4-3}Substitute a=3 and b=4
    \displaystyle =\displaystyle \dfrac{0-9}{4-3}Substitute f\left(3\right)=9 and f\left(4\right)=0
    \displaystyle =\displaystyle -9Evaluate the subtraction and division

    Between 3 seconds and the time the ball hits the ground, it is traveling an average of -9\text{ ft/s}.

b

Determine the interval(s) the ball is traveling at its fastest speed.

Worked Solution
Create a strategy

The speed is greatest when the absolute value of the rate of change of the height of the ball is greatest.

Apply the idea

From part (a), we found

  • The rate of change of the height of the ball in the interval between t=0 and t=1 was found to be 9\text{ ft/s}

  • The rate of change of the height of the ball in the interval between t=1 and t=2 was found to be 3\text{ ft/s}

  • The rate of change of the height of the ball in the interval between t=2 and t=3 was found to be -3\text{ ft/s}

  • The rate of change of the height of the ball in the interval between t=3 and t=4 was found to be -9\text{ ft/s}

The ball was traveling fastest between t=0 and t=1 and between t=3 and t=4.

Reflect and check

The rate of change for a function over a specific interval will be positive when the function is increasing and negative when the function is decreasing. In this context, the rate of change is positive when the height of the ball increases. The rate of change is negative when the height of the ball decreases over the time interval.

We were not concerned with the height of the ball, but rather the speed of the ball. This is why a rate of 9\text{ ft/s} is the same as a rate of -9\text{ ft/s}.

c

Classify the function based on its rate of change.

Worked Solution
Create a strategy

From part (a), we found the rate of change is variable. We can take a look at how the rates change over time to get a better idea of the function family this function belongs to.

Apply the idea

The function begins increasing, then increases at a slower rate until it reaches a maximum value, then decreases at that same rate, then decreases at an increasing rate which was the same rate at which it began increasing.

The pattern of symmetric increasing and decreasing rates means there will be a vertical line of symmetry through the x-value of the maximum point. These features are the characteristics of a quadratic function.

Reflect and check

The graph shown in the instructions is an upside down parabola. This is another feature that confirms the function is quadratic.

Idea summary

Constant, linear, and absolute value functions have a constant rate of change while quadratic and exponential functions have variable rates of change. Although the average rate of change for an exponential function varies, it grows or decays at a constant percent rate of change.

The rate of change, the structure of the equation, and the shape of the graph can help us classify the function into the correct family.

Outcomes

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.

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