1. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1 1 Section 3.2 Graphing Linear Equations Using Intercepts Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1

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3. 3 Linear Functions All equations of the form Ax + By = C are straight lines when graphed, as long as A and B are not both zero. Such an equation is called the standard form of the equation of the line. To graph equations of this form, two very important points are used – the intercepts.

4. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 4 An x-intercept of a graph is the x-coordinate of a point where the graph intersects the x-axis. The y-coordinate of the x- intercept is always zero. The graph of y = 4x – 8 crosses the x-axis at (2, 0) and that point is the x-intercept. x-intercept

5. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 5 The y-intercept of a graph is the y-coordinate of a point where the graph intersects the y-axis. The x-coordinate of the y-intercept is always zero. The graph of y = 3x + 4 crosses the y-axis at (0, 4) and that point is the y-intercept. y-intercept

6. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 6 6 Objective #1: Examples

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9. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 9 9 Objective #1: Examples

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11. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 11 x- and y-Intercepts Using Intercepts to Graph Ax + By = C 1) Find the x-intercept. Let y = 0 and solve for x. 2) Find the y-intercept. Let x = 0 and solve for y. 3) Find a checkpoint, a third ordered-pair solution. 4) Graph the equation by drawing a line through the three points.

12. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 12 Graphing Using InterceptsEXAMPLE SOLUTION Graph the equation using intercepts. 1) Find the x-intercept. Let y = 0 and then solve for x. Replace y with 0 Multiply and simplify

13. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 13 Graphing Using Intercepts 2) Find the y-intercept. Let x = 0 and then solve for y. Replace x with 0 Multiply and simplify Divide by 4 CONTINUED The y-intercept is 3, so the line passes through (0,3).

14. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 14 Graphing Using Intercepts 3) Find a checkpoint, a third ordered-pair solution. For our checkpoint, we will let x = 1 (because x = 1 is not the x- intercept) and find the corresponding value for y. Replace x with 1 Multiply Add 2 to both sides CONTINUED The checkpoint is the ordered pair (1, 3.5). Divide by 4 and simplify

15. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 15 Graphing Using Intercepts 4) Graph the equation by drawing a line through the three points. The three points in the figure below lie along the same line. Drawing a line through the three points results in the graph of. CONTINUED (1,3.5) (-6,0) (0,3)

16. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 16 Example Use intercepts for graphing 3x + 4y = 12 1.Find the x-intercept. Let y = 0 and solve for x. 3x + 4(0) = 12 3x = 12 x = 4 The x-intercept is 4. The line passes through (4, 0). 2.Find the y-intercept. Let x = 0 and solve for y. 3(0) + 4y = 12 4y = 12 y = 3 The y-intercept is 3. The line passes through (0, 3). Graphing Using Intercepts

17. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 17 Example Graph 3x + 4y = 12. 3.Find a checkpoint, a third ordered-pair solution. Let x = 2 3(2) + 4y = 12 6 + 4y = 12 4y = 6 y = 6/4 = 1.5 A checkpoint is the ordered pair (2, 1.5) Graphing Using InterceptsCONTINUED

18. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 18 Example : Graph 3x + 4y = 12. 4.Graph the equation by drawing a line through the intercepts and checkpoint. Graphing Using InterceptsCONTINUED

19. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 19 Copyright © 2013, 2009, 2006 Pearson Education, Inc. 19 Objective #2: Examples

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25. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 25 Copyright © 2013, 2009, 2006 Pearson Education, Inc. 25 Objective #2: ExamplesCONTINUED

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28. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 28 Copyright © 2013, 2009, 2006 Pearson Education, Inc. 28 Objective #2: ExamplesCONTINUED

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30. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 30 Horizontal and Vertical Lines Equation of a Horizontal Line (0,b) A horizontal line is given by an equation of the form y = b where b is the y- intercept. Equation of a Vertical Line (a,0) A vertical line is given by an equation of the form x = a where a is the x-intercept.

31. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 31 Horizontal and Vertical LinesEXAMPLE SOLUTION xy = -4(x,y)(x,y) -3-4(-3,-4) 1-4(1,-4) 6-4(6,-4) Upon plotting the three resultant points and connecting the points with a line, the graph to the right is the solution. (1,-4) (-3,-4)(6,-4)

32. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 32 Horizontal and Vertical LinesEXAMPLE SOLUTION Graph the linear equation: x = 5. x = 5y(x,y)(x,y) 5-3(5,-3) 51(5,1) 55(5,5) Upon plotting the three points and connecting the points with a line, the graph to the right is the solution. (5,1) (5,-3) (5,5)

33. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 33 Copyright © 2013, 2009, 2006 Pearson Education, Inc. 33 Objective #3: Examples

34. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 34 Copyright © 2013, 2009, 2006 Pearson Education, Inc. 34 Objective #3: Examples

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36. Copyright © 2013, 2009, 2006 Pearson Education, Inc. 36 Copyright © 2013, 2009, 2006 Pearson Education, Inc. 36 Objective #3: Examples