1. Chapter 6 Section 5 – Slide 1 Copyright © 2009 Pearson Education, Inc. AND

2. Copyright © 2009 Pearson Education, Inc. Chapter 6 Section 5 – Slide 2 Unit 4 Graphs

3. Copyright © 2009 Pearson Education, Inc. Chapter 6 Section 5 – Slide 3 WHAT YOU WILL LEARN Order of Operations Solving linear and quadratic equations, and linear inequalities in one variable Evaluating a formula Solving application problems involving linear, quadratic and exponential equations

4. Copyright © 2009 Pearson Education, Inc. Chapter 6 Section 5 – Slide 4 WHAT YOU WILL LEARN Solving application problems dealing with variation Graphing equations and functions, including linear, quadratic, and exponential equations

5. Copyright © 2009 Pearson Education, Inc. Chapter 6 Section 5 – Slide 5 Section 1 Variation

6. Chapter 6 Section 5 – Slide 6 Copyright © 2009 Pearson Education, Inc. Direct Variation Variation is an equation that relates one variable to one or more other variables. In direct variation, the values of the two related variables increase or decrease together. If a variable y varies directly with a variable x, then y = kx where k is the constant of proportionality (or the variation constant).

7. Chapter 6 Section 5 – Slide 7 Copyright © 2009 Pearson Education, Inc. Example The amount of interest earned on an investment, I, varies directly as the interest rate, r. If the interest earned is $50 when the interest rate is 5%, find the amount of interest earned when the interest rate is 7%. I = kr 50 = k(0.05) 1000 = k

8. Chapter 6 Section 5 – Slide 8 Copyright © 2009 Pearson Education, Inc. Example (continued) k = 1000, r = 7% I = kr I = 1000(0.07) I = 70 The amount of interest earned is $70.

9. Chapter 6 Section 5 – Slide 9 Copyright © 2009 Pearson Education, Inc. Inverse Variation When two quantities vary inversely, as one quantity increases, the other quantity decreases, and vice versa. If a variable y varies inversely with a variable, x, then where k is the constant of proportionality.

10. Chapter 6 Section 5 – Slide 10 Copyright © 2009 Pearson Education, Inc. Example Suppose y varies inversely as x. If y = 12 when x = 18, find y when x = 21. Now substitute 216 for k, and find y when x = 21.

11. Slide 6 - 11 Copyright © 2009 Pearson Education, Inc. For a constant speed, the distance traveled, d, varies directly as the elapsed time, t. If d = 50 miles when t = 40 minutes, find the distance traveled of an object with the same speed when the elapsed time is 100 minutes. a.125 miles b.80 miles c.100 miles d.150 miles

12. Slide 6 - 12 Copyright © 2009 Pearson Education, Inc. For a constant speed, the distance traveled, d, varies directly as the elapsed time, t. If d = 50 miles when t = 40 minutes, find the distance traveled of an object with the same speed when the elapsed time is 100 minutes. a.125 miles b.80 miles c.100 miles d.150 miles

13. Chapter 6 Section 5 – Slide 13 Copyright © 2009 Pearson Education, Inc. Joint Variation One quantity may vary directly as the product of two or more other quantities. The general form of a joint variation, where y, varies directly as x and z, is y = kxz where k is the constant of proportionality.

14. Chapter 6 Section 5 – Slide 14 Copyright © 2009 Pearson Education, Inc. Example The area, A, of a triangle varies jointly as its base, b, and height, h. If the area of a triangle is 48 in 2 when its base is 12 in. and its height is 8 in., find the area of a triangle whose base is 15 in. and whose height is 20 in.

15. Chapter 6 Section 5 – Slide 15 Copyright © 2009 Pearson Education, Inc. Combined Variation A varies jointly as B and C and inversely as the square of D. If A = 1 when B = 9, C = 4, and D = 6, find A when B = 8, C = 12, and D = 5. Write the equation.

16. Chapter 6 Section 5 – Slide 16 Copyright © 2009 Pearson Education, Inc. Combined Variation (continued) Find the constant of proportionality. Now find A.

17. Copyright © 2009 Pearson Education, Inc. Chapter 6 Section 5 – Slide 17 Section 2 Linear Inequalities

18. Chapter 6 Section 5 – Slide 18 Copyright © 2009 Pearson Education, Inc. Symbols of Inequality a < b means that a is less than b. a  b means that a is less than or equal to b. a > b means that a is greater than b. a  b means that a is greater than or equal to b. Find the solution to an inequality by adding, subtracting, multiplying or dividing both sides by the same number or expression. Change the direction of the inequality symbol when multiplying or dividing both sides of an inequality by a negative number.

19. Chapter 6 Section 5 – Slide 19 Copyright © 2009 Pearson Education, Inc. Example: Graphing Graph the solution set of x  4, where x is a real number, on the number line. The numbers less than or equal to 4 are all the points on the number line to the left of 4 and 4 itself. The closed circle at 4 shows that 4 is included in the solution set.

20. Chapter 6 Section 5 – Slide 20 Copyright © 2009 Pearson Education, Inc. Example: Graphing Graph the solution set of x > 3, where x is a real number, on the number line. The numbers greater than 3 are all the points on the number line to the right of 3. The open circle at 3 is used to indicate that 3 is not included in the solution set.

21. Chapter 6 Section 5 – Slide 21 Copyright © 2009 Pearson Education, Inc. Example: Solve and graph the solution Solve 3x – 8 < 10 and graph the solution set. The solution set is all real numbers less than 6.

22. Slide 6 - 22 Copyright © 2009 Pearson Education, Inc. Graph the solution set of –2x + 6 ≥ 4x – 10 on the real number line. a. c. b. d.

23. Slide 6 - 23 Copyright © 2009 Pearson Education, Inc. Graph the solution set of –2x + 6 ≥ 4x – 10 on the real number line. a. c. b. d.

24. Chapter 6 Section 5 – Slide 24 Copyright © 2009 Pearson Education, Inc. Compound Inequality Graph the solution set of the inequality  4 < x  3 where x is an integer. The solution set is the integers between  4 and 3, including 3.

25. Chapter 6 Section 5 – Slide 25 Copyright © 2009 Pearson Education, Inc. Compound Inequality (continued) Graph the solution set of the inequality  4 < x  3 where x is a real number The solution set consists of all real numbers between  4 and 3, including the 3 but not the  4.

26. Chapter 6 Section 5 – Slide 26 Copyright © 2009 Pearson Education, Inc. Example A student must have an average (the mean) on five tests that is greater than or equal to 85% but less than 92% to receive a final grade of B. Jamal’s scores on the first four tests were 98%, 89%, 88%, and 93%. What range of scores on the fifth test will give him a B in the course?

27. Chapter 6 Section 5 – Slide 27 Copyright © 2009 Pearson Education, Inc. Example (continued) Let x = Jamal’s score on the fifth test. Then: So Jama will receive a grade of B in the course if his score on the fifth test is greater than or equal to 57 and less than 92.

28. Copyright © 2009 Pearson Education, Inc. Chapter 6 Section 5 – Slide 28 Section 3 Graphing Linear Equations

29. Chapter 6 Section 5 – Slide 29 Copyright © 2009 Pearson Education, Inc. Rectangular Coordinate System x-axis y-axis origin Quadrant I Quadrant II Quadrant IIIQuadrant IV The horizontal line is called the x-axis. The vertical line is called the y-axis. The point of intersection is the origin.

30. Chapter 6 Section 5 – Slide 30 Copyright © 2009 Pearson Education, Inc. Plotting Points Each point in the xy-plane corresponds to a unique ordered pair (a, b). Plot the point (2, 4). Starting from the origin: Move 2 units right Move 4 units up 2 units 4 units

31. Chapter 6 Section 5 – Slide 31 Copyright © 2009 Pearson Education, Inc. Graphing Linear Equations Graph the equation y = 5x + 2 33 11 0  2/5 20 yx

32. Chapter 6 Section 5 – Slide 32 Copyright © 2009 Pearson Education, Inc. To Graph Equations by Plotting Points Solve the equation for y. Select at least three values for x and find their corresponding values of y. Plot the points. The points should be in a straight line. Draw a line through the set of points and place arrow tips at both ends of the line.

33. Chapter 6 Section 5 – Slide 33 Copyright © 2009 Pearson Education, Inc. Graphing Using Intercepts The x-intercept is found by letting y = 0 and solving for x. Example: y =  3x + 6 0 =  3x + 6  6 =  3x 2 = x The y-intercept is found by letting x = 0 and solving for y. Example: y =  3x + 6 y =  3(0) + 6 y = 6

34. Chapter 6 Section 5 – Slide 34 Copyright © 2009 Pearson Education, Inc. Example: Graph 3x + 2y = 6 Find the x-intercept. 3x + 2y = 6 3x + 2(0) = 6 3x = 6 x = 2 Find the y-intercept. 3x + 2y = 6 3(0) + 2y = 6 2y = 6 y = 3

35. Slide 6 - 35 Copyright © 2009 Pearson Education, Inc. Graph a. c. b. d. y x 8 4 y x y x y x 8 –4 4 8 –8

36. Slide 6 - 36 Copyright © 2009 Pearson Education, Inc. Graph a. c. b. d. y x 8 4 y x y x y x 8 –4 4 8 –8

37. Chapter 6 Section 5 – Slide 37 Copyright © 2009 Pearson Education, Inc. Slope The ratio of the vertical change to the horizontal change for any two points on the line.

38. Chapter 6 Section 5 – Slide 38 Copyright © 2009 Pearson Education, Inc. Types of Slope Positive slope rises from left to right. Negative slope falls from left to right. The slope of a vertical line is undefined. The slope of a horizontal line is zero. zero negative undefined positive

39. Chapter 6 Section 5 – Slide 39 Copyright © 2009 Pearson Education, Inc. Example: Finding Slope Find the slope of the line through the points (5,  3) and (  2,  3).

40. Chapter 6 Section 5 – Slide 40 Copyright © 2009 Pearson Education, Inc. The Slope-Intercept Form of a Line Slope-Intercept Form of the Equation of the Line y = mx + b where m is the slope of the line and (0, b) is the y-intercept of the line.

41. Chapter 6 Section 5 – Slide 41 Copyright © 2009 Pearson Education, Inc. Graphing Equations by Using the Slope and y-Intercept Solve the equation for y to place the equation in slope-intercept form. Determine the slope and y-intercept from the equation. Plot the y-intercept. Obtain a second point using the slope. Draw a straight line through the points.

42. Chapter 6 Section 5 – Slide 42 Copyright © 2009 Pearson Education, Inc. Example Graph 2x  3y = 9. Write in slope-intercept form. The y-intercept is (0,  3) and the slope is 2/3.

43. Chapter 6 Section 5 – Slide 43 Copyright © 2009 Pearson Education, Inc. Example continued Plot a point at (0,  3) on the y-axis, then move up 2 units and to the right 3 units.

44. Chapter 6 Section 5 – Slide 44 Copyright © 2009 Pearson Education, Inc. Horizontal Lines Graph y =  3. y is always equal to  3, the value of y can never be 0. The graph is parallel to the x-axis.

45. Chapter 6 Section 5 – Slide 45 Copyright © 2009 Pearson Education, Inc. Vertical Lines Graph x =  3. x always equals  3, the value of x can never be 0. The graph is parallel to the y-axis.