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Activities for Intermediate Algebra: Instructors’ Guide

Katherine Yoshiwara

Section 5.2 Graphs of Functions

In this Lesson we use function notation with graphs, construct the graph of a function, use the vertical line test, and solve equations and inequalities graphically.

Activity 5.2.1. The Graph of a Function.

The graph shows the U.S. unemployment rate as a function of time, \(U=f(t)\text{,}\) for the years 1985-2004.
  1. Evaluate \(f(1993)\text{,}\) and explain its meaning for this problem.
  2. When did the unemployment rate reach its highest value, and what was its highest value? Write your answer with function notation.
  3. When did the unemployment rate fall to its lowest value, and what was its lowest value? Write your answer with function notation.
  4. Solve the equation \(f(t)=4.5\text{,}\) and explain its meaning for this problem.
  5. Over what intervals was the unemployment rate increasing?

Activity 5.2.2. Functions Defined by Graphs.

The figure shows the graph of the function \(y=F(s)\text{.}\) A solid dot means that the point is part of the graph, and an open circle means that the point is not part of the graph.
  1. Find \(F(-3), F(-2)\) and \(F(2)\text{.}\)
  2. For what value(s) of \(s\) is \(F(s)=-1\) ?
  3. Find the maximum value of \(F(s)\text{.}\) For what value(s) of \(s\) does \(F\) take on its maximum value?
  4. Find the minimum value of \(F(s)\) For what value(s) of \(s\) does \(F\) take on its minimum value?

Activity 5.2.3. Solving Equations and Inequalities.

Each figure shows a graph of
\begin{equation*} B = f(p)= \dfrac{1}{3}p^3 - 3p + 2 \end{equation*}
Use the graph to solve the following equations and inequalities. Show your work on the graph. Write your answers in interval notation.
  1. \(\dfrac{1}{3}p^3 - 3p + 2 = 6\)
  2. \(\dfrac{1}{3}p^3 - 3p + 2 = 2\)
  3. \(\dfrac{1}{3}p^3 - 3p + 2 \lt 2\)

Subsection 5.2.1 Check Your Understanding

  1. In Activity 1, what is the input variable, and what is the output variable?
  2. In Activity 1d, how did you solve the equation?
  3. Is it possible for a function to have more than one maximum value? Is it possible for a function to take on its maximum value at more than one point?
  4. In Activity 3, how do you know that the graph represents a function?

Subsection 5.2.2 Wrap-Up

In this Lesson, we worked on the following skills and goals related to functions:
  • Use function notation to label points on a graph
  • Interpret the coordinates of a point on the graph of a function
  • Use the vertical line test
  • Solve an equation or inequality graphically

Subsection 5.2.3 Questions for Writing or Discussion

  1. Describe how to find the value of \(f(3)\) from a graph of \(f(3)\text{.}\)
  2. Explain how to construct the graph of a function from its equation.
  3. Explain how to use the vertical line test.
  4. Explain how to solve an inequality by using a graph.

Concept Questions.

  1. If \(f(8)=2\text{,}\) what point lies on the graph of \(f\text{?}\)
    1. \(\displaystyle (8,2)\)
    2. \(\displaystyle (2,8)\)
    3. Both (a) and (b).
    4. All points between 2 and 8.
  2. Write an equation that says \((8,17)\) lies on the graph of \(f\text{.}\)
    1. \(\displaystyle 8+17=g\)
    2. \(\displaystyle g+8=17\)
    3. \(\displaystyle g(8)=17\)
    4. \(\displaystyle 8=\dfrac{17}{g}\)
  3. Which of the following is true?
    1. It is not possible for the function to take on the same \(y\)-value at two different \(x\)-values.
    2. The maximum value of \(y=f(x)\) may occur at two different \(x\)-values.
    3. The maximum value of the function \(y=f(x)\) is the largest \(x\)-value that appears on the graph.
    4. \(f(x)=0\) at the \(x\)-intercept of \(y=f(x)\text{.}\)
  4. When is a function called decreasing?
    1. If its \(x\)-values increase from left to right.
    2. If its \(y\)-values decrease when its \(x\)-values decrease.
    3. If its \(y\)-values decrease when its \(x\)-values increase.
    4. If the graph lies below the \(x\)-axis.
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