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Section 1.4 Graphs of Functions

Subsection 1. New vocabulary

We can use function notation to describe a graph.

Fill in the blanks:

  1. The point \((a,b)\) lies on the graph of \(f\) if and only if .
  2. Each point on the graph of \(y=f(x)\) has coordinates .
  3. A graph of a function is increasing if the values get larger as we read from left to right.
  4. The maximum value of a function is the of the highest point on the graph.

Subsubsection Exercises

How do you know that the point \((1,9)\) lies on the graph of \(f(t)=-16t^2+20t+5\) ?

Answer

Because \(f(1)=9\text{.}\)

What are the coordinates of any point on the graph of \(h=f(t)\) ?

Answer

\((t, f(t))\)

Subsection 2. Solve equations and inequalities graphically

Subsubsection Examples

Every point on the graph of an equation \(y=f(x)\) tells us the solution of the equation for a particular value of \(y\text{.}\)

Example 1.63.

Use the graph of \(y=\dfrac{-2}{3}x-4\) to solve the equation

\begin{equation*} \dfrac{-2}{3}x-4=-2\text{.} \end{equation*}
Solution

We see that \(y\) has been replaced by \(-2\) in the equation for the graph. So we look for the point on the graph that has \(y\)-coordinate \(-2\text{.}\)

This point, labeled \(P\) on the graph at right, has \(x\)-coordinate \(-3\text{.}\) Because it lies on the graph, the point \(P(-3,-2)\) is a solution of the equation \(y=\dfrac{-2}{3}x-4\text{.}\)

line

But this statement also tells us that \(-3\) is a solution of the equation \(\dfrac{-2}{3}x-4=2\text{.}\) You can check that substituting \(x=-3\) into this equation produces a true statement.

Example 1.64.

Use the graph of \(y=\dfrac{-2}{3}x-4\) to solve the inequality

\begin{equation*} \dfrac{-2}{3}x-4 \le -2\text{.} \end{equation*}
Solution

We would like to find the \(x\)-coordinates of all points on the graph that have \(y\)-coordinate less than or equal to \(-2\text{.}\) These points on the graph are indicated by the heavy portion of the line.

The \(x\)-coordinates of these points are shown by the heavy portion of the \(x\)-axis. The solution is \(x \ge -3\text{,}\) or in interval notation, \([-3,\infty)\) .

line

Subsubsection Exercises

Use the graph to solve the equation or inequality. (Note the scales on the axes.) Show your solutions on the graph. Then verify your solutions by solving algebraically.

  1. \(\displaystyle -9-\dfrac{x}{4}=-2\)
  2. \(\displaystyle -9-\dfrac{x}{4} \ge -5\)
graph
Answer
  1. \(\displaystyle x =-28\)
  2. \(\displaystyle x \le -16\)

Use the graph to solve the equation or inequality. (Note the scales on the axes.) Show your solutions on the graph. Then verify your solutions by solving algebraically.

  1. \(\displaystyle 7-\dfrac{2x}{3}=3\)
  2. \(\displaystyle 7-\dfrac{2x}{3} \lt -1\)
graph
Answer
  1. \(\displaystyle x =6\)
  2. \(\displaystyle x \gt 12\)