Section 3.1: Math M111: Lecture Notes For Chapter 3 Note: Make sure you already printed the graphing papers Plotting Points, Quadrant s signs, x-intercepts and y-intercepts Example 1: Plot the following points A(2, 3) ; B(-2, 2), C(-3, -2), D(3,-1) Example 2: Plot the following points E(0, 4) ; F(0, -1), G(4, 0), H(-4, 0) Graphing Linear Equations (straight lines) Linear equation can be written in two forms: Standard Form: Ax + By = C (the constant number is isolated) Slope-intercept Form: y = mx + b (y is isolated) Example 3: graph 4x - 5y = 20 Example 4: graph 3x + 2y = 12 Example 5: graph 2x - 3y = 0 Example 6: graph 2y - 12 = 0 Example 7: graph x 6 = 0 Find the x- and y- intercepts. Then graph each equation (page 146 in the book) #36: graph x 2 y = -4 #38: graph 5 6 x + y = 2 7 7 #42: graph x = -3 #44: graph y + 2 = 0 #48: graph 4 y = 3x Note: You can use Graphmatica to graph all homework problems: x + 1 should be written as: abs(x+1) y = x 2-3x should be written as: y = x^2-3x y = x 4 should be written as: y =sqrt(x - 4) y 3 = x 3 should be written as: y =3 / (x-3) 1
Section 3.2: Slope of a line Slope of a line = Vertical Changes = Horizontal Changes Diff.in y Diff.in x Example 1: Find the slope of the line passing the points (-2, 4) and (3, -1) Example 2: Find the slope of the line passing the points (2,-3) and (-1, 3) m > 0 or positive slope, then the line is increasing or rising m < 0 or negative slope, then the line is decreasing or falling m = 0, then the line is horizontal m = undefined, no slope, then the line is vertical Slope-Intercept Equation y = mx + b (m is the slope, b is the y-intercept) Example 3: Graph the following: 4x y = 4 and find the slope. Find the slope of the line and sketch the graph (page 158) : #30: x + 3y = -6 #36: y + 5 = 0 Example 4: x + 3 = 0 Graphing using the slope and one point #38: Through (-4, 2), m = 1/2 #40: Through (0, -2), m = -2/3 #44: Through (2, -5), m = 0 #46: Through (-4, 1), m = undefined. Two lines are parallel if they have same slope but different y-intercepts Example 5: 3x y = -4 and 3x y = 2 where the slope is = 3 in both Example 6: The line passing (12, 2) and (10, 0) and the line passing (4, 6) and (8, 10) have same slope Two line are perpendicular if the slopes are m and 1/m (The product = -1). Example 7: y = 2x - 5 and y= -1/2 x + 3 Example 8: Determine weather the given pair are parallel a) y + 7x = -9 and -3y = 21x + 7 b) y + 8 = -6x and -2x + y = 5 Example 9: Determine weather the given pair are perpendicular a) y = -x + 7 and y = x + 3 b) 2x 5y = -3 and 2x + 5y = 4 2
Section 3.3: Linear Equation of Two Variables Remember: Standard Form: Ax + By = C (the constant number is isolated) Slope-intercept Form: y = mx + b (y is isolated) Example 1: graph y = 2x + 1 Example 2: Match each equation with the graph that it most closely resembels: 1) y = -2/3 x + 4 ; graph: A B 2) y = 2x + 4 ; graph: 3) y = -5/2 x - 2 ; graph: C D 4) y = 4/3 x - 2 ; graph: 5) y = -2x ; graph: E F 6) y = 3x ; graph: 7) y = 2 ; graph: G H 8) x= 3 ; graph: Example 3: Graph using the slope and the y-intercepts: a) y = 2/5 x + 3 b) -2x + 3y = 6 c) 2x + 3y = 6 d) -2x - 5y = 10 3
Finding the equation of a line, y = mx + b Case 1: One point is given and the slope: Example 4: Find the equation of the line having the given slope and containing the given point and write your answer in both forms (Standard and slop-intercept): a) m = -2 ; (2, 8) b) m = -4/5 ; (2, 3) Case 2: Two points are given: Example 5 Find the equation of the line containing the given pair of points and write your answer in both forms (Standard and slop-intercept): a) (-4, -7) and (-2, -1) b) (0, -5) and (3, 0) Special Cases: Example 6 Find the equation of the line that satisfies the given condition: a) Through (-6, -2) and m = 0 (horizontal) b) Through (-2, 3) and m = undefined (vertical) Write an equation of the line containing the given point and parallel to the given line: a) (-4, -5) ; 2x + y = -3 b) (-7, 0) ; 2y + 5x= 6 Write an equation of the line containing the given point and perpendicular to the given line: a) (4, 1) ; x - 3y = 9 b) (-3, -4) ; 6y - 3x= 2 c) (-2, 5) ; x = 4 4
Section 3.4: Linear Inequalities in Two variables Example 1: graph 2x + 3y > 1 Example 2: graph 2x + 3y > 1 Example 3: graph 3x - y < 3 Example 4: graph y > 3x and compare it to: y > 3x, y < 3x, y = 3x Section 3.5: Functions Domain (x-axis or input): Independent Range (y-axis or output): Dependent Example 1: y = f(x) = 5x - 3 (y is a function of x, y depends on x) x = 0, then y = -3 or f(0) = -3 Function: (Vertical line crosses only once) profit profit profit year year year A B C Graph A: A function, vertical line crosses only once. Different input, Different output. Graph B: A function, vertical line crosses only once. Different input, Same output (the profit in two different years were the same). Graph C: Not a function, vertical line crosses more than once. Same input, Different output (two different profits for the same year). Tell whether each relation defines a function 6. {(8, 0), (5, 4), (9, 3), (3, 8)} 8. {(9, -2), (-3, 5), (9, 2)} 10. {(-12, 5), (-10, 3), (8, 3)} 5
Decide whether each relation defines a function and give the domain and range. 12. {(2, 5), (3, 7), (4, 9), (5, 11)} The following exercises will be solved in class: # 14, 16, 18, 20, 22 Domain: find the set of all possible inputs that would not cause a zero in the denominator or negative under an even root Example 2: Find the domain of a) y 2 = x 4 b) 5 x y = c) x 1 + x y = 3 d) y = x + 2 x Decide whether each relation defines y as a function of x. Give the domain. 24. y = x 3 26. x = y 4 28. y = 6 x + 8 30. x y < 3 32. y = x 34. xy = 3 36. y = 9 2x 38. 7 y = x 16 Let f ( x) = 3x + 4 and g ( x) = x 2 + 4x + 1. Find the following: 42. f ( 3) 44. g (10) 46. g (k) 48. g( x) 50. f ( a + 4) 6