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Section 1.3 Equations and Graphs

Subsection Anatomy of a Graph

We use equations to express the relationship between two variabes. A graph is a way of visualizing an equation.

A graph has two axes, horizontal and vertical, and the values for the variables are displayed along the axes. The first, or input, variable is displayed on the horizontal axis. The second, or output, variable is displayed on the vertical axis.

The graph itself shows how the two variables are related.

Example 1.17.

The graph below shows the relationship between Delbert's age, \(D\text{,}\) and Francine's age, \(F\text{.}\) Write an equation for Francine's age in terms of Delbert's age.

Relationship between Delbert's and Francine's ages
Solution

In this graph, \(D\) is the input variable and \(F\) is the output variable. We make a table of values by reading points on the graph.

\(D\) \(0\) \(2\) \(\alert{3}\) \(4\) \(6\) \(9\)
\(F\) \(4\) \(6\) \(\alert{7}\) \(8\) \(10\) \(13\)
From the table, we see that the value of \(F\) is always 4 more than the value of \(D\text{.}\) Thus, Francine is exactly four years older than Delbert, or \(F=D+4\text{.}\)

Reading Questions Reading Questions

1.

A is a way of visualizing an algebraic equation.

2.

The values of the variables are displayed on the .

3.

The input variable is located on the axis.

Each point on a graph has two coordinates, which designate the position of the point. For example, the point labeled \(P\) in Example 1.17 has horizontal coordinate 3 and vertical coordinate 7.

In Example 1.17 above, we write the coordinates of point \(P\) inside parentheses as an ordered pair: \((3,7)\text{.}\) The order of the coordinates makes a difference. We always list the horizontal coordinate first, then the vertical coordinate.

Reading Questions Reading Questions

4.

The position of a point on the graph is given by its .

5.

The notation \((x,y)\) is called .

6.

In an ordered pair, we always list the coordinate first.

Look Closer.

How does the graph illustrate the equation \(F=D+4\text{?}\) The coordinates of each point on the graph are values for \(D\) and \(F\) that make the equation true. The coordinates of the point \(P\text{,}\) namely \(D=\alert{3}\) and \(F=\alert{7}\text{,}\) represent the fact that when Delbert was 3 years old, Francine was 7 years old. If we substitute these values into our equation we get

\begin{equation*} \begin{aligned} F \amp = D+4\\ \alert{7} \amp = \alert{3}+4 \end{aligned} \end{equation*}

which is true.

An ordered pair that makes an equation true is called a solution of the equation. Each point on the graph represents a solution of the equation.

Exercise 1.18.
  1. Locate on the graph each of the ordered pairs listed in the table above, and make a dot there. Label each point with its coordinates.
  2. By substituting its coordinates into the equation, verify that each point you labeled in part (a) represents a solution of the equation \(F=D+4\text{.}\)

Reading Questions Reading Questions

7.

An ordered pair whose coordinates make the equation true is called a of the equation.

Subsection Graphing an Equation

The simplest way to draw a graph for an equation is to make a table of values and plot the points.

Example 1.19.

Graph the equation \(y=8-x\text{.}\)

Solution

In this equation, \(x\) is the input variable and \(y\) is the output variable. We choose several values for \(x\) and use the equation to find the corresponding values for \(y\text{.}\)

\(x\) Calculation \(y\) \((x,y)\)
\(0\) \(y=8-0=8\) \(8\) \((0,8)\)
\(2\) \(y=8-2=6\) \(6\) \((2,6)\)
\(5\) \(y=8-5=3\) \(3\) \((5,3)\)
\(8\) \(y=8-8=0\) \(0\) \((8,0)\)
points on line
Each ordered pair \((x,y)\) represents a point on the graph of the equation. We plot the points on the grid and connect them with a smooth curve, as shown above.

Subsection Choosing Scales for the Axes

If the variables in an equation have very large (or very small) values, we must choose scales for the axes that fit these values.

To graph an equation.
  1. Make a table of values.
  2. Choose scales for the axes.
  3. Plot the points and connect them with a smooth curve.
Example 1.20.

Graph the equation \(y=250+x\text{.}\)

Solution

To graph this equation, we choose multiples of 50 for the \(x\)-values.

\(x\) Calculation \(y\) \((x,y)\)
\(0\) \(y=250+0=250\) \(250\) \((0,250)\)
\(50\) \(y=250+50=300\) \(300\) \((50, 300)\)
\(100\) \(y=250+100=350\) \(350\) \((100, 350)\)
\(200\) \(y=250+200=450\) \(450\) \((200, 450)\)
graph of line

The largest \(y\)-value in the table is 450, so we scale the axis to a little larger than 450, say, 500. We plot the ordered pairs to obtain the graph shown above.

Look Closer.

How do we know what scales to use on the axes? For most graphs, it is best to have between ten and twenty tick marks on each axis, or the graph will be hard to read. We choose intervals of convenient size for the particular problem, such as 5, 10, 25, 100, or 1000. It is not necessary to use the same scale on both axes.

Caution 1.21.

It is important that the scales on the axes be evenly spaced. Each tick mark must represent the same interval. The scales on the graph below are incorrectly labeled.

\(\alert{\text{Incorrect!}~~~ \rightarrow}\)

inappropriately labeled grid

Subsection Solving Equations with Graphs

In Example 1 we showed a graph of the equation

\begin{equation*} F=D+4 \end{equation*}

which gives Francine's age, \(F\text{,}\) in terms of Delbert's age, \(D\text{.}\) We can use the graph to answer two types of questions about the equation \(F=D+4\text{:}\)

  1. Given a value of \(D\text{,}\) find the corresponding value of \(F\text{.}\)
  2. Given a value of \(F\text{,}\) find the corresponding value of \(D\text{.}\)

The first of these tasks is another way of evaluating the algebraic expression \(D+4\text{,}\) and the second task is another way of solving an equation.

Example 1.22.
  1. Use the graph of the equation \(F=D+4\) to evaluate the expression \(D+4\) for \(D=7\text{.}\) Verify your answer algebraically.
  2. Use the graph of the equation \(F=D+4\) to solve the equation \(13=D+4\text{.}\) Verify your answer algebraically.
Solution
  1. We locate \(D=7\) on the horizontal axis, as shown at left below. Then we move vertically to point on the graph with \(D\)-coordinate 7. Finally, we move horizontally from point \(A\) to the vertical axis to find the \(F\)-coordinate. The coordinates of point \(A\) are \((7,11)\text{,}\) which tells us that when \(D=7,~F=11\text{.}\) Our answer is 11.

    To verify the answer algebraically, we evaluate the expression for \(D=7\text{:}\) we substitute \(\alert{7}\) for \(D\) into the equation:

    \begin{equation*} F=D+4=\alert{7}+4=11 \end{equation*}
  2. We locate \(F=\alert{13}\) on the vertical axis. We move horizontally to point \(B\) on the graph shown above right, with \(F\)-coordinate 13. From point \(B\text{,}\) we move vertically to the horizontal axis to find the \(D\)-coordinate. The coordinates of point \(B\) are \((9,13)\text{,}\) which tells us that when \(F=13,~D=9\text{.}\) Our answer is 9.

    To verify the answer algebraically, we can substitute \(D=9\) into the equation:

    \begin{equation*} F=D+4=\alert{9}+4=13 \end{equation*}
Example 1.23.

Here is a graph of

\begin{equation*} y=1.5x \end{equation*}

Use the graph to solve the equation \(1.5x=3.75\text{.}\)

y=1.5x
Solution

By comparing the equation of the graph with the equation we want to solve, we see that \(y\) has been replaced by \(\alert{3.75}\text{.}\)

\begin{equation*} \begin{aligned} y \amp = 1.5x\\ \downarrow \amp \\ \alert{3.75} \amp = 1.5x \end{aligned} \end{equation*}

We locate 3.75 on the \(y\)-axis, then find the point on the graph with \(y\)-coordinate 3.75.

y=1.5x
The \(x\)-coordinate of this point is 2.5, so the ordered pair \((2.5, 3.75)\) is a solution of the equation \(y=1.5x\text{,}\) and \(x=2.5\) is the solution of the equation \(1.5x=3.75\text{.}\)

Reading Questions Reading Questions

8.

The graph of an equation is a picture of its .

9.

To solve an equation using a graph, we start on the axis.

Subsection Skills Warm-Up

Exercises Exercises

In Exercises 1–2, what value corresponds to each labeled point?

1.
numberline
2.
number line
3.
  1. Label the axis.
    number line
  2. On the axis in part (a), plot 1400 and 8350.
4.
  1. Label the axis below with 16 tick marks (not counting zero), the highest being 800.
    number line
  2. On the axis in part (a), plot 132 and 614.
5.
  1. Label an axis in increments of 40,000 from 0 to 600,000.
  2. On your axis, plot 250,000 and 472,600.
6.
  1. Label an axis in increments of 0.5 from 0 to 5.
  2. On the axis in part (a), plot 1.3 and 3.77.

Solutions Answers to Skills Warm-Up

Exercises Exercises

Exercises Homework 1.3

For Problems 1–2, decide whether the ordered pairs are solutions of the equation whose graph is shown.

1.
  1. \((6.5,3)\)
  2. \((0,3.5)\)
  3. \((8,2)\)
  4. \((4.5,1)\)
2.
  1. \((2,6)\)
  2. \((4,2)\)
  3. \((10,0)\)
  4. \((11,7)\)

For Problems 3–6, decide whether the ordered pairs are solutions of the given equation.

3.

\(y=\dfrac{3}{4}x\)

  1. \((8,6)\)
  2. \((12,16)\)
  3. \((2,3)\)
  4. \((6,\dfrac{9}{2})\)
4.

\(y=\dfrac{x}{2.5}\)

  1. \((1,2.5)\)
  2. \((25,10)\)
  3. \((5,2)\)
  4. \((8,20)\)
5.

\(w=z-1.8\)

  1. \((10, 8.8)\)
  2. \((6,7.8)\)
  3. \((2,\dfrac{1}{5})\)
  4. \((9.2,7.4)\)
6.

\(w=120-z\)

  1. \((0, 120)\)
  2. \((65,55)\)
  3. \((150,30)\)
  4. \((9.6,2.4)\)

For Problems 7–8, state the interval that each grid line represents on the horizontal and vertical axes.

7.
grid
8.
grid

For the graphs in Problems 9–10,

  1. Fill in the table with the correct coordinates. Choose your own points to complete the table.
  2. Look for a pattern in your table, and write an equation for the second variable in terms of the first variable.
9.
\(x\) \(\hphantom{00}\) \(2\) \(5\) \(\hphantom{00}\) \(10\) \(\hphantom{00}\) \(\hphantom{00}\)
\(y\) \(16\) \(\hphantom{00}\) \(\hphantom{00}\) \(10\) \(\hphantom{00}\) \(\hphantom{00}\) \(\hphantom{00}\)
line on grid
10.
\(x\) \(0\) \(1\) \(\hphantom{00}\) \(\hphantom{00}\) \(7\) \(\hphantom{00}\) \(\hphantom{00}\)
\(y\) \(\hphantom{00}\) \(\hphantom{00}\) \(16\) \(20\) \(\hphantom{00}\) \(\hphantom{00}\) \(\hphantom{00}\)
line on grid

For Problems 11–14, choose the equation that describes the graph. (A graph might have more than one equation.)

  1. \(y=150-x\)
  2. \(y=15-x\)
  3. \(y=0.2x\)
  4. \(y=\dfrac{x}{60}\)
  5. \(y=\dfrac{x}{5}\)
  6. \(y=15+x\)
  7. \(y=\dfrac{60}{x}\)
  8. \(y=5x\)
11.
line
12.
curve
13.
line
14.
line

Use the three steps in the Lesson to make the graphs in Problems 15–17.

15.

The college Bookstore charges a 25% markup over wholesale prices. If the wholesale price of a book is \(w\) dollars, the bookstore's price \(p\) is given by

\begin{equation*} p=1.25w \end{equation*}
  1. How much does the bookstore charge for a book whose wholesale price is $28?
  2. Evaluate the formula to find the bookstore's price for various books.
    \(w\) \(12\) \(16\) \(20\) \(30\) \(36\) \(40\)
    \(p\) \(\hphantom{00}\) \(\hphantom{00}\) \(\hphantom{00}\) \(\hphantom{00}\) \(\hphantom{00}\) \(\hphantom{00}\)
  3. Plot the values from the table and graph the equation.
    grid
16.

Uncle Ray's diet allows him to eat a total of 1000 calories for lunch and dinner. Let \(L\) stand for the number of calories in Uncle Ray's lunch, and \(D\) for the number of calories in his dinner.

  1. Write an equation for \(D\) in terms of \(L\text{.}\)
  2. Complete the table.
    \(L\) \(250\) \(300\) \(450\) \(60\)
    \(D\) \(\hphantom{00}\) \(\hphantom{00}\) \(\hphantom{00}\) \(\hphantom{00}\)
  3. Label the scales on the axes and graph the equation. Use intervals of 100 on both axes.
    grid
17.

The Green Co-op plans to divide its profit of $3600 from the sale of environmentally safe products among its members. Let \(m\) stand for the number of members, and \(S\) for the share of the profits each member will receive.

  1. Write an equation for \(S\) in terms of \(m\text{.}\)
  2. Complete the table.
    \(m\) \(20\) \(40\) \(60\) \(100\)
    \(S\) \(\hphantom{00}\) \(\hphantom{00}\) \(\hphantom{00}\) \(\hphantom{00}\)
  3. Label the scales on the axes and graph the equation. Use intervals of 20 on the vertical axis.
    grid

For Problems 18–19, solve the equation graphically by drawing on the graph. (You may have to estimate some of your solutions.) Then verify your answer by solving algebraically.

18.

The graph of \(k=p-20\) is shown below. Use the graph to solve the equations:

  1. \(90=p-20\)
  2. \(p-20=40\)
line
19.

The graph of \(d=18g\) is shown below. Use the graph to solve the equations:

  1. \(250=18g\)
  2. \(18g=210\)
line

For Problems 20–21,

  1. Give the coordinates of the indicated point.
  2. Explain what those coordinates tell us about the problem situation.
20.

The graph gives the sales tax, \(T\text{,}\) in terms of the price \(p\) of an item, where the amounts are in dollars.

line and point
21.

The graph gives the weight, \(W\text{,}\) in ounces of a chocolate bunny that is \(h\) inches tall.

curve and point

For Problems 22–25, use the formulas in Section 1.2.

22.
  1. Suppose your father loans you $2000 to be repaid with 4% annual interest when you finish school. Write an equation for the amount of interest \(I\) you will owe after \(t\) years.
  2. How much interest will you owe if you finish school in 2 years? In 4 years? In 7 years? Make a table showing your answers as ordered pairs.
  3. Use your table to choose appropriate scales for the axes, then plot points and sketch a graph of the equation
23.
  1. The Earth Alliance made $6000 in revenue from selling tickets to Earth Day, an event for children. Write an equation for its profit \(P\) in terms of its costs.
  2. What was their profit if their costs were $800? $1000? $2500? Make a table showing your answers as ordered pairs.
  3. Use your table to choose appropriate scales for the axes, then plot points and sketch a graph of the equation.
24.
  1. Edgar's great-aunt plans to put $5000 in a trust for Edgar until he turns 21, three years from now. She has a choice of several different accounts. Write an equation for the amount of interest the money will earn in 3 years if the account pays interest rate \(r\text{.}\)
  2. How much interest will the $5000 earn in an account that pays 8% interest? 10% interest? 12% interest? Make a table showing your answers as ordered pairs.
  3. Use your table to choose appropriate scales for the axes, then plot points and sketch a graph of the equation.
25.
  1. Delbert keeps track of the total number of points he earns on homework assignments, each of which is worth 30 points. At the end of the semester he has 540 points. Write an equation for Delbert's average homework score in terms of the number of assignments, \(n\text{.}\)
  2. What is Delbert's average score if there were 20 assignments? 25 assignments? 30 assignments? Make a table showing your answers as ordered pairs.
  3. Use your table to choose appropriate scales for the axes, then plot points and sketch a graph of the equation.