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§ 1.3 Intercepts

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**Intercepts Example: Find the x-intercept of 4 = x – 3y Let y = 0.**

Then 4 = x – 3y becomes 4 = x – 3(0) Replace y with 0. 4 = x Simplify the right side. So the x-intercept is (4,0).

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**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.

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**Graph by Plotting Intercepts**

Example: Graph the linear equation 4 = x – 3y by plotting intercepts. We previously found that the y-intercept is (0, ) and the x-intercept is (4, 0). Along with the intercepts, for the third solution, let y = 1. 4 = x – 3(1) Replace y with 1. 4 = x – 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.

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**Graph by Plotting Intercepts**

x y Example continued: Plot the two intercepts (0, ) and (4, 0) along with the third solution (7, 1). (4, 0) (7, 1) (0, ) Draw the line that contains the three points.

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**Graph by Plotting Intercepts**

Example: Graph 2x = 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 2x = 0 x = 0, so the x-intercept is (0,0) Notice that the x- and y-intercept are the same point. Continued.

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**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 2x = 4 x = 2, so another point is (2, 4) Continued.

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**Graph by Plotting Intercepts**

x y Example continued: (3, 6) (0, 0) (2, 4) Plot all three of the solutions (0, 0), (3, 6) and (2, 4). Draw the line that contains the three points.

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**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).

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**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.

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**Graph by Plotting Intercepts**

x y (0, 3) Example continued: y = 3

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**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.

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**Graph by Plotting Intercepts**

x y (– 3, 0) Example continued: x = – 3

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