171S1.1 Graphs, Functions, Models August 23, 2012

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1 MAT 171 D10 8:00 9:15 MAT 171 D11 9:30 10:45 Aug 21 7:11 AM Aug 21 7:14 AM MAT 171 D14 2:00 3:15 MAT 171 D15 3:30 4:45 Aug 21 7:14 AM Aug 21 7:14 AM 1

2 MAT 171 Dr. Claude Moore, CFCC CHAPTER 1: Graphs, Functions, and Models 1.1 Introduction to Graphing 1.2 Functions and Graphs 1.3 Linear Functions, Slope, and Applications 1.4 Equations of Lines and Modeling 1.5 Linear Equations, Functions, Zeros and Applications 1.6 Solving Linear Inequalities Session 1 introduces the Course, CourseCompass, and Chapter 1: Graphs, Functions, and Models. This session is available in CourseCompass. Read the Announcements to find Session 1. I suggest that you view the examples that were not worked in your section. If you have questions or need assistance, please contact me or go to the Math Lab (S606) or the Learning Lab (L Building). If the CFCC website is not available, you may access the following with links: mycfcc : WebAdvisor: Click on the globe icon to access a free graphing tool. Follow the instructions given in the file. 1.1 Introduction to Graphing 1. Plot points. 2. Determine whether an ordered pair is a solution of an equation. 3. Find the x and y intercepts of an equation of the form Ax + By = C. 4. Graph equations. 5. Find the distance between two points in the plane and find the midpoint of a segment. 6. Find an equation of a circle with a given center and radius, and given an equation of a circle in standard form, find the center and the radius. 7. Graph equations of circles. Cartesian Coordinate System To graph or plot a point, the first coordinate tells us to move left or right from the origin. The second coordinate tells us to move up or down. ( 3, 5) Plot ( 3, 5). Move 3 units left. Next, we move 5 units up. Plot the point. 2

3 Solutions of Equations Equations in two variables have solutions (x, y) that are ordered pairs. : 2x + 3y = 18 When an ordered pair is substituted into the equation, the result is a true equation. The ordered pair has to be a solution of the equation to receive a true statement. s a. Determine whether the ordered pair ( 5, 7) is a solution of 2x + 3y = 18. 2( 5) + 3(7)? ? = 18 FALSE ( 5, 7) is not a solution. b. Determine whether the ordered pair (3, 4) is a solution of 2x + 3y = 18. 2(3) + 3(4)? ? = 18 TRUE (3, 4) is a solution. Graphs of Equations To graph an equation is to make a drawing that represents the solutions of that equation. Click on the globe icon to access a free graphing tool. Follow the instructions given in the file. x Intercept The point at which the graph crosses the x axis. An x intercept is a point (a, 0). To find a, let y = 0 and solve for x. : Find the x intercept of 2x + 3y = 18. 2x + 3(0) = 18 2x = 18 x = 9 The x intercept is (9, 0). 3

4 y Intercept The point at which the graph crosses the y axis. A y intercept is a point (0, b). To find b, let x = 0 and solve for y. : Find the y intercept of 2x + 3y = 18. 2(0) + 3y = 18 3y = 18 y = 6 The y intercept is (0, 6). Graph 2x + 3y We already found the x intercept: (9, 0) We already found the y intercept: (0, 6) We find a third solution as a check. If x is replaced with 5, then Thus, is a (continued) Graph: 2x + 3y = 18. x intercept: (9, 0) y intercept: (0, 6) Third point: Graph y = x 2 9x 12. 4

5 The Distance Formula The distance d between any two points (x 1, y 1 ) and (x 2, y 2 ) is given by Find the distance between the points ( 2, 2) and (3, 6). Midpoint Formula If the endpoints of a segment are (x 1, y 1 ) and (x 2, y 2 ), then the coordinates of the midpoint are Find the midpoint of a segment whose endpoints are ( 4, 2) and (2, 5). 5

6 Circles A circle is the set of all points in a plane that are a fixed distance r from a center (h, k). The equation of a circle with center (h, k) and radius r, in standard form, is (x h) 2 + (y k) 2 = r 2. Find an equation of a circle having radius 5 and center (3, 7). Using the standard form, we have (x h) 2 + (y k) 2 = r 2 [x 3] 2 + [y ( 7)] 2 = 5 2 (x 3) 2 + (y + 7) 2 = /22. Find the intercepts and graph the line: 3y + 2x = 6 70/10. Use substitution to determine whether the given ordered pairs are solutions of the given equation. Jan 5 7:57 PM Jan 5 8:01 PM 6

7 71/58. Use a graphing calculator to graph the equation in the standard window: y = x 2 5x /70. Find the distance between the pair of points. Give an exact answer and, where appropriate, an approximation to three decimal places. Jan 5 8:01 PM Jan 5 8:03 PM 71/77. The points ( 3, 1) and (9, 4) are the endpoints of the diameter of a circle. Find the length of the radius of the circle. 72/92. Find the midpoint of the segment having the given endpoints. 72/112. Find the center and the radius of the circle. Then graph the circle by hand. Check your graph with a graphing calculator. (x 7) 2 + (y + 2) 2 = 25 Jan 5 8:05 PM Jan 5 8:07 PM 7