3 3 Intercepts Intercepts of axes where graph

§ 3. 3 Intercepts

Intercepts of axes (where graph crosses the axes) Since all points on the x-axis have a y-coordinate of 0, to find the x-intercept, let y = 0 and solve for x. Since all points on the y-axis have an x-coordinate of 0, to find the y-intercept, let x = 0 and solve for y. Martin-Gay, Beginning and Intermediate Algebra, 4 ed 2

Intercepts Example: Find the y-intercept of 4 = x – 3 y Let x = 0. Then 4 = x – 3 y becomes 4 = 0 – 3 y Replace x with 0. 4 = – 3 y Simplify the right side. =y Divide both sides by – 3. So the y-intercept is (0, ). Martin-Gay, Beginning and Intermediate Algebra, 4 ed 3

Intercepts Example: Find the x-intercept of 4 = x – 3 y Let y = 0. Then 4 = x – 3 y becomes 4 = x – 3(0) Replace y with 0. 4=x Simplify the right side. So the x-intercept is (4, 0). Martin-Gay, Beginning and Intermediate Algebra, 4 ed 4

Graph by Plotting Intercepts Example: Graph the linear equation 4 = x – 3 y by plotting intercepts. We previously found that the y-intercept is (0, x-intercept is (4, 0). ) and the Along with the intercepts, for the third solution, let y = 1. 4 = x – 3(1) Replace y with 1. 4=x– 3 Simplify the right side. 4+3=x Add 3 to both sides. 7=x Simplify the left side. So the third solution is (7, 1). Continued. Martin-Gay, Beginning and Intermediate Algebra, 4 ed 5

Graph by Plotting Intercepts y Example continued: Plot the two intercepts (0, and (4, 0) along with the third solution (7, 1). ) (4, 0) Draw the line that contains the three points. (0, ) Martin-Gay, Beginning and Intermediate Algebra, 4 ed (7, 1) x 6

Graph by Plotting Intercepts Example: Graph 2 x = y by plotting intercepts To find the y-intercept, let x = 0 2(0) = y 0 = y, so the y-intercept is (0, 0) To find the x-intercept, let y = 0 2 x = 0, so the x-intercept is (0, 0) Notice that the x- and y-intercept are the same point. Continued. Martin-Gay, Beginning and Intermediate Algebra, 4 ed 7

Graph by Plotting Intercepts Example continued: Find two other solutions to the equation. Let x = 3 2(3) = y 6 = y, so another point is (3, 6) Let y = 4 2 x = 4 x = 2, so another point is (2, 4) Continued. Martin-Gay, Beginning and Intermediate Algebra, 4 ed 8

Graph by Plotting Intercepts Example continued: y (3, 6) Plot all three of the solutions (0, 0), (3, 6) and (2, 4). Draw the line that contains the three points. (2, 4) (0, 0) Martin-Gay, Beginning and Intermediate Algebra, 4 ed x 9

Vertical and Horizontal Lines Vertical Lines The graph of x = c, where c is a real number is a vertical line with x-intercept (c, 0). Horizontal Lines The graph of y = c, where c is a real number, is a horizontal line with y-intercept (0, c). Martin-Gay, Beginning and Intermediate Algebra, 4 ed 10

Graph by Plotting Intercepts Example: Graph y = 3 Note that this line can be written as y = 0·x + 3 The y-intercept is (0, 3), but there is no x-intercept. (Since an x-intercept would be found by letting y = 0, and 0 0·x + 3, there is no x-intercept. ) Every value we substitute for x gives a y-coordinate of 3. The graph will be a horizontal line through the point (0, 3) on the y-axis. Continued. Martin-Gay, Beginning and Intermediate Algebra, 4 ed 11

Graph by Plotting Intercepts y Example continued: y=3 (0, 3) x Martin-Gay, Beginning and Intermediate Algebra, 4 ed 12

Graph by Plotting Intercepts Example: Graph x = – 3. This equation can be written x = 0·y – 3. When y = 0, x = – 3, so the x-intercept is (– 3, 0), but there is no y-intercept. Any value we substitute for y gives an x-coordinate of – 3. So the graph will be a vertical line through the point (– 3, 0) on the x-axis. Continued. Martin-Gay, Beginning and Intermediate Algebra, 4 ed 13

Graph by Plotting Intercepts y Example continued: x=– 3 (– 3, 0) Martin-Gay, Beginning and Intermediate Algebra, 4 ed x 14