# Equations and Their Graphs

Objectives
After reviewing this unit, you will be able to:

• Draw a graph from a given equation.
• Determine whether a given point lies on the graph of a given equation.

Graphing an Equation
In section one we introduced an example using the cost of pizza based on the number of toppings on the pizza. In this example, a plain pizza with no toppings was priced at 7 dollars. As you add one topping, the cost goes up by 75 cents. (If you wish to review the example, return to the first unit.) We found the equation that relates the total price of a pizza to the number of toppings on the pizza to be y = 7.00 + .75x. Now let's look at how you can draw a graph that illustrates this relationship. The procedure for doing this is:

1. Generate a list of points for the relationship.
2. Draw a set of axes and define the scale.
3. Plot the points on the axes.
4. Draw the line by connecting the points.

Now let's work through these steps with our pizza example:

1. 1. Generate a list of points for the relationship.
The first step in the process is to generate a list of points to graph. You do this by selecting several values for thex coordinate. You should always make sure you find at least three points to graph. (NOTE: Using three points serves as an extra check that you have done all the calculations correctly. All three should lie on the same straight line.) Once you have selected at least three values for x, use the equation to calculate their correspondingy values. While you are doing this you should display the information in a table.

In the pizza example, the equation is y = 7.00 + .75x. You first select values ofx you will solve for. You then substitute these values into the equation and solve for they values. Your list of points may be kept in a table like the one below. For our pizza example, a table may start out this way:

x
Number of Toppings
y
Final Cost
0 7.00
1 7.75
2
3
4

In the table below, for each givenx value you can see the calculation of each of they values.

 x y Number of Toppings Final Cost Let x = 0:        y = 7.00 + .75 (0) y = 7.00 + 0 y = 7.00 0 7.00 Let x = 1:         y = 7.00 + .75 (1) y = 7.00 + .75 y = 7.75 1 7.75 Let x = 2:         y = 7.00 + .75 (2) y = 7.00 + 1.50 y = 8.50 2 8.50 Let x = 3:         y = 7.00 + .75 (3) y = 7.00 + 2.25 y = 9.25 3 9.25 Let x = 4:         y = 7.00 + .75 (4) y = 7.00 + 3.00 y = 10.00 4 10.00

After you have completed your table, you should end up with the following list of points:

(0, 7.00) , (1, 7.75), (2, 8.50), (3, 9.25), (4, 10.00)

Notice they are written using the (x, y) notation. You are now ready to create the graph of these points on a set of axes.

1. 2. Draw a set of axes and define the scale.

 Once you have your list of points you are ready to plot them on a graph. The first step in drawing the graph is setting up the axes and determining the scale. The points you have to plot are: (0, 7.00), (1, 7.75), (2, 8.50), (3, 9.25), (4, 10.00) Notice that thexvalues range from 0 to 4 and theyvalues go from 7 to 10. The scale of the two axes must include all the points. Below is a set of axes drawn to do just that. Notice that the distance between the points must be equal on each axis but does not have to be the same for both axes. The x-axis goes up to 5 on this diagram, and the y-axis up to 10; the scale on each axis can be different.

1. 3. Plot the points on the axes.

 After you have drawn the axes, you are ready to plot the points. (How to plot points was discussed in the first part of this book. If you feel you need to review this, reread the discussion on plotting points in the second unit.) Below we plot the points on a set of axes.

1. 4. Draw the line by connecting the points.

 Once you have plotted each of the points, you can connect them and draw a straight line.

Checking a Point in the Equation
If, by chance, you have a point and you wish to determine if it lies on the line, you simply go through the same process as generating points. Use the x value given in the point and insert it into the equation. Compare the y value calculated with the one given in the point.

Example

Does point (6, 10) lie on the line y = 7.00 + .75 x given in our pizza example? To determine this, we need to plug the point (6, 10) into the equation.

 x y Number of Toppings Final Cost Let x = 6 y = 7.00 + .75 (6) y = 7.00 + 4.50 y = 11.50 6 11.50

As we can see here, the point with an x value of 6 that does lie on the line is (6, 11.5). This means that the point (6, 10) does not lie on our line.

You are now ready to try graphing an equation. Move on to the first practice for this unit. If you have already completed the first problem try the additional pratice before moving on.