PreCalculus

9.3. Equations in two variables

A graph on the Cartesian plane represents the set of all the ordered pairs of x and y that satisfy an equation. To draw the graph of an equation, we can plot points and sketch a curve through the points. First, create a table of ordered pairs by choosing values of either x or y and computing the corresponding values of the other variable from the equation. Then plot the points given by these ordered pairs.

For example, sketch the graph of: y = (x + 3)/2

Figure 9-2: Plotting points

Creating a table or ordered pairs

A computer program can be used to quickly calculate a list of ordered pairs of points for a complicated curve. For example, create a list of ordered pairs for the equation:

     x + 1
y = ———————
     x - 1

First note that at x = 1, the denominator on the right side is equal to zero. We can write a simple program in the Zeno programming language to compute y for several values of x except x = 1 as follows:

program

    var x : int
    var y : real

    put " x", tab(10), " y"
    put "-------------------"

    for x := -5...5 do
        if x = 1 then
            put x:2, tab(10), "undefined"
        else
            y := (x + 1)/(x - 1)
            put x:2, tab(10), y:9:6
        end if
    end for

end program

This will print out the list:

 x         y
-------------------
-5         0.666667
-4         0.600000
-3         0.500000
-2         0.333333
-1         0.000000
 0        -1.000000
 1        undefined
 2         3.000000
 3         2.000000
 4         1.666667
 5         1.500000

Intercepts

The point where a graph crosses a coordinate axis is called an intercept. The crossing point on the x-axis is called the x-intercept; the crossing point on the y-axis is the y-intercept. Since the x-axis crosses the y-axis at y = 0, we can find the x-intercept by setting y = 0. Similarly, we can find the y-intercept by setting x = 0.

Example, find the intercepts for: y = x² - x - 2

Set y = 0:  0 = x² - x - 2 = (x + 1)(x - 2)
The x-intercepts are -1, 2.

Set x = 0:  y = 0² - 0 - 2 = -2
The y-intercept is -2.