# Graphing Equations

You know what they say: " a picture is worth a thousand numbers". Given an equation in x and y, its graph consists of all the points in the Cartesian coordinate system whose coordinates (x,y) are solutions to the equation.

Consider the equation: . Let us find a few ordered pairs (x,y) solutions of the equation and plot them in the xy-coordinate system. Just like in Example 4, we find a solution by setting one variable, say x, to a specific value and solving for the variable y.  The following table summarizes our findings when we set x to -3, -2, -1, 0, 1, 2, and 3.

 x y -3 0 -2 -2 -1 -1 0 -3 0 3 1 1 2 2 3 0

This table displays 12 ordered pairs solutions of the equation:

(-3,0), (-2,  ), (-2,  ), (-1,  ), (-1,  ), (0,-3), (0,3), (1,  ), (1,  ), (2,  ), (2,  ), and (3,0).

Plotting these solutions on the Cartesian coordinate system, we get:

Figure 1.

Of course, this is not the complete graph of the equation since we only plotted a few solutions. The complete graph can only be obtained if we plotted all the possible solutions; which is impossible for us since there are infinitely many solutions. However, the few points that we plotted reveal a definite pattern in the way the solutions lay on the Cartesian plane. An examination of the plot suggests that, if all solutions are plotted, the resulting graph is the circle centered at the origin and of radius 3 (see figure below):

Figure 2.

The fact that, based on a few (solutions) points on the graph, we were able to project the overall shape of the graph is a tribute to the power of visualization. We relate much better to pictures than to text, numbers, or equations. The equation  represents a very familiar shape: that of a circle.

Example 5

Is the point (1,2) on the graph of the equation ?

Solution

If we substitute 1 for x and 2 for y into the equation we get: . That is 1 = 5, which is a false statement. Therefore the ordered pair (1,2) is not a solution of the equation and the point (1,2) is not on its graph.

Exploration 1

Refer to the interactive exploration below.

This interactive exploration is designed to graph equations in x and y.

You may enter any equation in x and y by entering the left hand side of the equation on the text-field that is left to the "=" sign and the right hand side of the equation on the text-field that is right to the "=" sign. Once you enter an equation:

(i)                  Using what we did in Example 4, calculate a few ordered pairs solutions to the equation (use pencil and paper)

(ii)                Enter the ordered pairs, one at a time, in the text-field next to the button "plot point" (enter the ordered pair without the parentheses) and then push the button "plot point". This will plot the point and verify numerically whether or not it is on the graph of the equation. Do that for all the ordered pairs you calculated from the previous step.

(iii)               Try to guess the pattern that the points you plotted follow and then check you guess by pushing the button "graph equation".

Notes On Entering And Graphing Equations

(i)                  When entering equations, remember to use * for multiplication, / for division, ^ for exponentiation, sqrt( ) for square root of, abs( ) for absolute value of.

(ii)                 If a point or the graph is not visible on the Cartesian coordinate system, try adjusting the viewing window so that the point or the graph is visible.

(iii)               Use the button "clear" to clear the Cartesian coordinate system when you are done with each equation.

Here are a few equations you might want to try:

(a)          [for left side, enter 2*x+3*y; for right side, enter 5]

(b)               [for left side, enter x^2+y^2; for right side, enter 9]

(c)           [for left side, enter abs(2*x)+abs(y); for right side, enter 4]

(d)          [left side: x^2/2+abs(2*y); right side: 5]

(e)       [left side: sqrt(1+x^2)+abs(y);  right side: 4]

(f)          [left side: 5*x+y/10; right side:  x*(x^2+y^2)]

The following figure was obtained by simultaneously graphing the following four equations:

(i)

(ii)

(iii)

(iv)

Figure 3.

Experiment with graphing equations on the exploration page and try to generate the most spectacular graph.

Intercepts

Given an equation in x and y. The x-intercepts are the points where the graph of the equation crosses the x-axis. The y-intercepts are the points where the graph of the equation crosses the y-axis.

For a point to be an x-intercept, it must be on the graph (which means it is a solution of the equation) and on the x-axis (which means that its ordinate (y-coordinate) is 0). This suggests the following strategy for finding the x-intercepts: to find the x-intercepts, substitute 0 for y into the equation and solve for x.

For a point to be a y-intercept, it must be on the graph (which means it is a solution of the equation) and on the y-axis (which means that its abscissa (x-coordinate) is 0). This suggests the following strategy for finding the y-intercepts: to find the y-intercepts, substitute 0 for x into the equation and solve for y.

Example 6

Find the x-intercepts and the y-intercepts for the following equation:

Solution

(v)                x-intercepts: substituting 0 for y, the equation becomes .

Solving for x, we get: x  = -3 or x  = 3. So there are two x-intercepts:

One at x = -3 and one at x = 3 (see Figure 4).

(vi)              y-intercepts: substituting 0 for x, the equation becomes  , or  (dividing both sides by 3)  . Solving for y, we get:

or                .

There are two y-intercepts (see Figure 4).

Figure 4.