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3.3 Intercepts

Identifying Intercepts The graph of y = 4x – 8 is shown below. Notice that this graph crosses the y-axis at the point (0, –8). This point is called the y-intercept. Likewise the graph crosses the x-axis at (2, 0). This point is called the x-intercept. The intercepts are (2, 0) and (0, –8).

Helpful Hint If a graph crosses the x-axis at (2, 0) and the y-axis at (0, –8), then (2, 0) (0, –8) Notice that for the x-intercept, the y-value is 0 and for the y-intercept, the x-value is 0. Note: Sometimes in mathematics, you may see just the number –8 stated as the y-intercept and 2 stated as the x-intercept. x-intercept y-intercept

Example Identify the x- and y-intercepts. a. b. x-intercept: (-1, 0), (3, 0) y-intercept: (0, -3) x-intercept: (2, 0)

Intercepts Finding x- and y-Intercepts To find the x-intercept, let y = 0 and solve for x. To find the y-intercept, let x = 0 and solve for y.

Example Graph 4 = x – 3y by finding and plotting its intercepts. To find the y-intercept, let x = 0. 4 = x – 3y 4 = 0 – 3y Replace x with 0. 4 = – 3y Simplify. = y Divide both sides by –3. Continued

Example (cont) To find the x-intercept, let y = 0. 4 = x – 3y 4 = x – 3(0) Replace y with 0. 4 = x Simplify. So the x-intercept is (4,0). Continued

Example (cont) Along with the intercepts, for the third solution, let y = 1. 4 = x – 3y 4 = x – 3(1) Replace y with 1. 4 = x – 3 Simplify. 4 + 3 = x Add 3 to both sides. 7 = x Simplify. So the third solution is (7, 1). Continued

Graph by Plotting Intercepts x y Now we plot the two intercepts and (4, 0) along with the third solution (7, 1). (4, 0) (7, 1) (0, ) The graph of 4 = x – 3y is the line drawn through these points.

Example Graph 2x = y by finding and plotting its 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). It’s the same point. What do we do? Continued

Helpful Hint Notice that any time (0, 0) is a point of a graph, then it is an x-intercept and a y-intercept. Why? It is the only point that lies on both axes.

Example (cont) Since we need at least 2 points to graph a line, we will have to find at least one more point. 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

Example (cont) x y (3, 6) (0, 0) (2, 4) Now we plot the three solutions (0, 0), (3, 6) and (2, 4). The graph of 2x = y is the line drawn through these points.

Example Graph: x = – 3 This equation can be written x = 0·y – 3. Any value we substitute for y gives an x-coordinate of –3. The graph is a vertical line with x-intercept –3, and no y-intercept. Continued

Example (cont) x y (-3, 0) The graph of x = –3.

Example Graph: y = 3 The equation can be written as y = 0·x + 3. For any x-value chosen, y is 3. The graph is a horizontal line with y-intercept 3, and no x-intercept . Continued

Example (cont) x y (0, 3) The graph of y = 3.

Vertical and Horizontal Lines x y 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). x y