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B. Using Equations to Represent Relations

As you reviewed in the Warm Up, relations can be represented in many different ways. There are advantages and disadvantages of each method. For example, a graph is a very visual representation of a relation, but it takes longer to produce than some of the other representations.

As you saw in the Investigation, relations can be represented by equations. Equations are a very concise way of representing relations and they can be algebraically manipulated quite easily.

To determine the ordered pairs that satisfy an equation, substitute values for one variable into the equation and solve for the other variable.

 

Example 1

Determine some points on the graph of the relation «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo»=«/mo»«mn»2«/mn»«mi»x«/mi»«mo»+«/mo»«mn»3«/mn»«/math». Use the points to sketch the graph of «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo»=«/mo»«mn»2«/mn»«mi»x«/mi»«mo»+«/mo»«mn»3«/mn»«/math».

The x-values you choose will depend on which part of the graph you are interested in representing. This becomes more important when you know the context of the relation. When there is no context, x-values near zero are often chosen.

Step 1: Begin by selecting some x-values. Here, integer values from –3 to 3 are used.

Step 2: Substitute each x-value into the equation to determine the corresponding y-value.

Step 3: Enter this y-value into the table.

Step 4: Repeat the same procedure to determine the other y-values and list each set as ordered pairs.

Step 5: Plot the points on a graph and draw a line through them.

For relations such as Â«math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo»=«/mo»«mn»2«/mn»«mi»x«/mi»«mo»+«/mo»«mn»3«/mn»«/math» where there is no context, we assume that all domain and range values are Real Numbers and a continuous line can be drawn to connect the points.