Graphs of Equations
Using a coordinate system with the horizontal axis labeled with the letter x and the vertical axis labeled with the letter y allows graphing of equations involving the letters x and y. We sometimes call the x-coordinate the input for the equation and the y-coordinate the output of the equation. To graph an equation means to graph all points (x,y) for which the equation is true.
Example 15. For the equation y = 5 - 2x, find y when x is 2.
Answer:
Replacing x with the number 2 gives the result y = 5 - 2(2) or y = 1. In other words, the point (2,1) is on the graph of the equation y = 5 - 2x.
Example 16. For the equation 5x - 2y = 40, find y when x = 10.
Answer:
Replacing x with 10 gives 5(10) - 2y = 40. Solving for y gives 50 - 40 = 2y or y = 5. In other words, the point (10,5) is on the graph of the equation 5x - 2y = 40.
By replacing the x value and the y value of an ordered pair in the equation, we can determine if the point satisfies the equation. To satisfy an equation is to find a solution of the equation. If the x and y values of the point are a solution to the equation, then the point is on the graph of the equation.
Example 17. For 3x + 2y = 12, determine if the point (2,3) satisfies the equation.
Answer:
Replacing x with 2 and y with 3 gives 3(2) + 2(3) = 12. Simplifying gives 6 + 6 =12. Since 12 = 12 is true, the point (2,3) satisfies the equation 3x + 2y = 12, and it is on the graph of the equation.
Example 18. For 3x + 2y = 12, determine if the point (1,-3) satisfies the equation.
Answer:
Replacing x with 1 and y with -3 gives 3(1) + 2(-3) = 12. Simplifying gives 3 - 6 = 12. Since -3 ≠ 12, the point (1,-3) does not satisfy the equation. Therefore, the point (1,-3) is not on the graph of the equation.
Example 19. For 
, determine if the point (5,10) satisfies the equation.
Answer:
Replacing x with 5 and y with 10 gives 
. Simplifying gives 
, 
, 
. Since 5 = 5 is true, the point (5,10) satisfies the equation.
The way to graph an equation is
a) select some x values
b) use these x values and the equation to find the y values
c) plot the points you have found
d) draw a line through the plotted points
It may be more convenient in some cases to select some y values first.
For this course, all of the equations we graph will be straight lines. However, not all graphs are straight lines. For instance, the graph of y = x2 is U shaped and is called a parabola.
Two convenient points to use for graphing a line are the intercepts of the line. The y-intercept is found by replacing x with 0 and finding y. The x-intercept is found by replacing y with 0 and finding x. The y-intercept is the point on the y-axis where the line crosses, and the x-intercept is the point on the x-axis where the line crosses.
Example 20. Find the intercepts of the line 3x + 8y = 48.
Answer:
Replacing x with 0 yields 3(0) + 8y = 48. So y = 6 and the y-intercept is the point (0,6). Replacing y with 0 yields 3x + 8(0) = 48. So x = 16 and the x-intercept is the point (16,0).
Example 21. Graph the equation y = 3x - 4 on the axes below.
Answer:
Replacing x with 0 in the equation gives y = -4. Replacing x with 2 gives y = 2. Replacing x with 3 gives y = 5. Thus, we must graph the corresponding points (0,-4), (2,2), and (3,5) and then draw a line through them. The x values must be selected intelligently so that the resulting points are not located out of the region covered by your coordinate axes. The three points and the line through them are shown below:
Example 22. Graph the equation y = x.
Answer:
This means to graph every point whose y-coordinate has the same value as its x-coordinate. Such a graph would consist of points such as (-3,-3) , (0,0), (1,1), (4,4) and all other points where the y-coordinate is equal to the x-coordinate. The graph is shown below:
Example 23. Graph the equation y = 2x - 1.
Answer:
This means to graph those points whose y-coordinate is 1 less than twice the x-coordinate. A few points on this graph would be (0,-1), (3,5), (100,199), and so on. The graph is shown below:
Example 24. Graph the equation 
.
.
Answer:
This means to graph those points whose y-coordinate is 1 more than two-thirds the x-coordinate. A good idea would be to plot the point where x = 0 and y = 1 and then points whose x values are multiples of three. For example, when x = 3, y = 3; when x = -3, y = -1; when x = 6, y = 5. Plot the points (0,1), (3,3), (-3,-1), and (6,5). The graph is shown below:
Example 25. Graph the equation 
.
Answer:
When x = 0, y = 1. Therefore, the y-intercept is the point (0,1). When y = 0, x = 4. Therefore, the x-intercept is (4,0), The graph is shown below:
Example 26. Graph the equation 
. This equation means that y can be any value, but x is always 4. Therefore, a few points on this line are (4,0), (4,-2), (4,2), and (4,5).
Example 27. Graph the equation 
. This equation means that x can be any value, but y is always 4. Therefore, a few points on this line are (0,4), (-2,4), (2,4), and (5,4).
