Intro. To Graphing Linear Equations
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1 Intro. To Graphing Linear Equations The Coordinate Plane A. The coordinate plane has 4 quadrants. B. Each point in the coordinate plain has an x-coordinate (the abscissa) and a y-coordinate (the ordinate). The point is stated as an ordered pair (x,y). C. Horizontal Axis is the X Axis. (y = 0) D. Vertical Axis is the Y- Axis (x = 0) Plot the following points: a) (3,7) b) (-4,5) c) (-6,-1) d) (6,-7) e) (5,0) f) (0,5) g) (-5,0) f) (0, -5) y-axis x-axis 1
2 Slope Intercept Form Before graphing linear equations, we need to be familiar with slope intercept form. To understand slope intercept form, we need to understand two major terms: The slope and the y-intercept. Slope (m): The slope measures the steepness of a non-vertical line. It is sometimes referred to as the rise over run. It s how fast and in what direction y changes compared to x. y-intercept: The y-intercept is where a line passes through the y axis. It is always stated as an ordered pair (x,y). The x coordinate is always zero. The y coordinate can be found by plugging in 0 for the X in the equation or by finding exactly where the line crosses the y-axis. What are the coordinates of the y-intercept line pictured in the diagram above? : Some of you have worked with slope intercept form of a linear equation before. You may remember: y = mx + b Using y = mx + b, can you figure out the equation of the line pictured above?: 2
3 Graphing Linear Equations Graphing The Linear Equation: y = 3x - 5 1) Find the slope: m = 3 m = 3. = y. 1 x 2) Find the y-intercept: x = 0, b = -5 (0, -5) 3) Plot the y-intercept 4) Use slope to find the next point: Start at (0,-5) m = 3. = y. up 3 on the y-axis 1 x right 1 on the x-axis (1,-2) Repeat: (2,1) (3,4) (4,7) 5) To plot to the left side of the y-axis, go to y-int. and do the opposite. (Down 3 on the y, left 1 on the x) (-1,-8) 6) Connect the dots. Do Now: 1) y = 2x + 1 2) y = -4x + 5 3
4 3) y = ½ x 3 4) y= - ⅔x + 2 5) y = -x 3 6) y= 5x 4
5 Q3 Quiz 1 Review Find the equation in slope intercept form and graph: If your printer doesn t print the graphs, you must use your own graph paper. 1) y = 4x - 6 2) y = -2x + 7 5
6 Find the equation in slope intercept form and graph: If your printer doesn t print the graphs, you must use your own graph paper. 3) y = -x - 5 4) y = 5x + 5 6
7 Find the equation in slope intercept form and graph: If your printer doesn t print the graphs, you must use your own graph paper. 5) y = - ½ x - 7 6) y = ⅗x - 4 7
8 Find the equation in slope intercept form and graph: If your printer doesn t print the graphs, you must use your own graph paper. 7) y = ⅔x 8) y = - ⅓x + 4 8
9 Finding the equation of a line in slope intercept form (y=mx + b) Example: Using slope intercept form [y = mx + b] Find the equation in slope intercept form of the line formed by (1,2) and (-2, -7). A. Find the slope (m): B. Use m and one point to find b: m = y 2 y 1 y = mx + b x 2 x 1 m= 3 x= 1 y= 2 m = (-7) (2). 2 = 3(1) + b (-2) (1) 2 = 3 + b -3-3 m = = b -3 m= 3 y = 3x 1 Example: Using point slope form [ y y 1 = m(x x 1 ) ] Find the equation in slope intercept form of the line formed by (1,2) and (-2, -7). A. Find the slope (m): B. Use m and one point to find b: m = y 2 y 1 y y 1 = m(x x 1 ) x 2 x 1 m= 3 x= 1 y= 2 y (2) = 3(x (1)) m = (-7) (2). y 2 = 3x - 3 (-2) (1) m = -9. y = 3x 1-3 m= 3 9
10 Find the equation in slope intercept form of the line formed by the given points. When you re finished, graph the equation on the give graph. 1) (4,-6) and (-8, 3) 10
11 2) (4,-3) and (9,-3) 3) (7,-2) and (7, 4) III. Special Slopes A. Zero Slope B. No Slope (undefined slope) * No change in Y * No change in X * Equation will be Y = * Equation will be X = * Horizontal Line * Vertical Line 11
12 Point-Slope Form y y 1 = m(x x 1 ) Slope Intercept Form y = mx + b y is by itself Standard Form: Ax + By = C Constant (number) is by itself Given the slope and 1 point, write the equation of the line in: (a) point-slope form, (b) slope intercept form, and (c) standard form: Example: m = ½ ; (-6,-1) a) Point-Slope Form b) Slope intercept form c) Standard Form 1) m = -2; (-3,1) a) Point-Slope Form b) Slope intercept form c) Standard Form 12
13 2) m = - ¾ ; (-8, 5) Point-Slope Form b) Slope intercept form c) Standard Form 3) m = ⅔; (-6, -4) Point-Slope Form b) Slope intercept form c) Standard Form 4) m = -1 (5, -1) Point-Slope Form b) Slope intercept form c) Standard Form 13
14 Find equation in slope intercept form and graph: 1) (3,-2)(-6,-8) 2) (-6,10) (9,-10) 3) (3,7) (3,-7) 4) (7,-6)(-3,4) 14
15 5) (5,-9)(-5,-9) 6) m= 4 (-2,-5) 7) m= ⅔ (-6,-7) 8) m= - (8,-1) 15
16 9) m = 0 (4,3) 10) m = undefined (-6, 5) 11) 16x -4y =36 12) 8x+24y = 96 16
17 13) y+7=2(x+1) 14) y+5=(2/5)(x+10) 15) y-7= ¾ (x-12) 16) y-2=-3(x-2) 17
18 Q3 Quiz 2 Review Find the equation in slope intercept form and graph: If your printer doesn t print the graphs, you must use your own graph paper. 1) y - 2 = -3(x 1) 2) 14x + 21y =
19 Find the equation in slope intercept form and graph: If your printer doesn t print the graphs, you must use your own graph paper. 3) y + 10 = 5(x + 2) 4) y 7 = ¼ (x 20) 19
20 Find the equation in slope intercept form and graph: If your printer doesn t print the graphs, you must use your own graph paper. 5) 8x 8y = 56 6) y + 6 = -1(x 4) 20
21 Find the equation in slope intercept form and graph: If your printer doesn t print the graphs, you must use your own graph paper. 7) 18x 12y = -12 8) y 15 = (-5/3)(x + 9) Answers: 1) y = -3x + 5 2) y = - ⅔ x - 4 3) y = 5x 4) y = ¼ x + 2 5) y = x - 7 6) y = - x 3 7) y = (3/2)x - 1 8) y = -(5/3)x 21
22 Graph both of the lines on the same set of axis: y = -2x + 6 y = -2x 5 IV. Parallel and Perpendicular Lines: A. Parallel Lines * Do not intersect * Have same slopes For the given line, find a line that is parallel and passes through the given point and graph Given Line: Parallel: Given Line: Parallel: 7) y = ⅓ x + 4 (6,1) 8) y = 4x 5 (2,13) Given Line: Parallel: Given Line: Parallel: 9) y = -⅔ x + 2 (-9,2) 10) y = 5x + 6 (4,-27) 22
23 Practice Problems: a) Use the two points to find the equation of the line. b) For the line found in part a, find a line that is parallel and passes through the given point. c) Graph both lines on the same set of axis. Given Line: Parallel: 1) (-5, 13) (3, -3) (4,-10) Given Line: Parallel: 2) (-6,0) (3,6) (6,3) 23
24 Given Line: Parallel: 3) (2,6)(-3,-19) (5,30) Given Line: Parallel: 4) (-4,3) (-8,6) (-4, 10) 24
25 Given Line: Parallel: 5) (2,-5) (-2, -5) (8,-2) Given Line: Parallel: 6) (-9,-11)(6,9) (-3,-9) 25
26 Given Line: Parallel: 7) (8,-3) (-4,9) (-2, 1) Given Line: Parallel: 8) (3,6)(3,-6) (7,-3) 26
27 Given Line: Parallel: 9) (4,-3)(-6,-8) (6,7) Given Line: Parallel: 10) (2,4)(-6,-12) (-3,-5) 27
28 11) Find the equation of the line parallel to y = 3x 2, passing through (-2, 1). 12) Find the equation of the line parallel to y = -½x 5, passing through (-2, 7) 13) Find the equation of the line parallel to y = -¼ x + 2, passing through (-8, 4) 14) Find the equation of the line parallel to y = (3/2)x + 6, passing through (-6, -11) 15) Find the equation of the line parallel to y = -5, passing through (2,7) 16) Find the equation of the line parallel to x = 5, passing through (6, -4). 28
29 Q3 Quiz 3 Review FOLLOW REQUIRED FORMAT AND SHOW ALL PROPER WORK! a) Use the two points to find the equation of the line. b) For the line found in part a, find a line that is parallel and passes through the given point. c) Graph both lines on the same set of axis. Given Line: Parallel: 1) (-4, 13) (3, -8) (4,-17) Given Line: Parallel: 2) (8,1) (-4,-5) (-6,2) 29
30 Given Line: Parallel: 3) (5,4) (-4,4) (-6,-7) For # s 4-7, just find the equation. You do not have to graph. 4) Find the equation of the line parallel to y = -⅗x 2, passing through (-5, 7). 5) Find the equation of the line parallel to y = 4x 5, passing through (-4, 9) 6) Find the equation of the line parallel to y = 2, passing through (-8, -9) 7) Find the equation of the line parallel to x = 5, passing through (-6, -11) 30
31 Solving Systems of Equations Graphically A system of equations is a collection of two or more equations with a same set of unknowns. In solving a system of equations, we try to find values for each of the unknowns that will satisfy every equation in the system. When solving a system containing two linear equations there will be one ordered pair (x,y) that will work in both equations. To solve such a system graphically, we will graph both lines on the same set of axis and look for the point of intersection. The point of intersection will be the one ordered pair that works in both equations. We must then CHECK the solution by substituting the x and y coordinates in BOTH ORIGINAL EQUATIONS. 1) Solve the following system graphically: y = 2x 5 y = - ⅓x
32 Solve each of the systems of equations graphically: 2) y + 1 = -3(x 1) 7x + 7y = 42 3) y 9 = ¾ (x 12) 6x + 12y =
33 4) 12x 8y = 48 y 4 = -2(x 2) 5) y + 5 = 2(x + 4) y 10 = - ½ (x + 4) 33
34 Solve each system graphically and check: 6) y = -4x -5 y = 2x -7 7) 6x + 3y =21 12x + 16y =
35 8) 12x 6y = -6 16x -8y = 40 9) y= -4 x = 7 35
36 10) y-2= (3/5)(x-10) y+11 =2(x+7) 11) 6x + 9y = 45 9x +15y = 75 36
37 12) x = 5 y-12 = -3(x+2) 13) 9x 18y = 126 y = -4 37
38 Q3 Quiz 4 Review 1) y = 2x + 6 y = - ½x - 4 2) 15x - 15y = -45 y - 2 = -3(x + 1) 38
39 3) y + 3 = (-2/3)(x - 3) y + 1 = -1(x + 2) 4) 24x - 18y = -18 y = -7 39
40 5) y + 2 = (-3/4)(x - 8) 17x - 34y = 204 6) x = -6 y + 15 = (5/3)(x + 9) 40
41 7) y - 5 = ¼(x - 4) 45x - 15y = 105 8) 24x - 12y = -72 y 2 = 2(x + 2) 41
42 9) 11x + 44y = -176 y - 3 = - ¼ (x + 4) 10) y + 4 = (2/3)(x + 12) 25x + 50y = -150 Answer Key: 1) (-4,-2) 2) (-1,2) 3) (-6,3) 4) (-6, -7) 5) (8,-2) 6) (-6,-10) 7) (4,5) 8) Many Solutions 9) No Solution 10) (-6,0) 42
43 Graphing Inequalities When we solved and graphed inequalities with only one variable (ex: x > 3), we moved on to compound inequalities (AND/OR). We would graph both inequalities on the same number line and decide what to keep based on whether it was an AND or an OR problem. When we graphed linear equations on the coordinate plane we moved on to solving systems of equations graphically. When we graph inequalities in two variables on the coordinate plane, we do not graph compound inequalities. We move on to solving systems of inequalities. It takes a little from both inequalities with one variable and solving systems graphically. Graph the Inequality: y > ¼ x + 3 Step 1: Graph the line. y > ¼ x + 3 m = ¼ = y = up 1 x r 4 y-int= (0,3) Step 2: Test a point one up from the from the y-int and one down from the y-int): (0, 2) (0, 4) 2 > ¼ (0)+3 4 > ¼ (0) > 3 4 > 3 FALSE TRUE Step 3: Shade towards the true point (0,4) When you test, you must do it in the original inequality! 43
44 1) 6x - 9y > 36 2) y - 3 > -2(x + 1) 44
45 3) 12x + 9y < 27 4) y + 4 > -3(x - 3) 45
46 5) y > 4 6) x < -6 46
47 Q3 Quiz 5 Review 1) 72x 216y < ) y + 1 > ⅖ (x + 10) 47
48 3) y 5 < - ½ (x + 10) 4) 48x + 12y <
49 5) x > 7 6) y < -2 49
50 7) x < -4 8) y > 6 50
51 Graphing Systems of Inequalities Solve the system of inequalities graphically: y > ¼ x + 3 y < 3x 5 Step 1: Graph the 1 st inequality Step 2: Graph the 2 nd inequality y > ¼ x + 3 y < 3x 5 m = ¼ = m = 3/1 = y-int= (0,3) y-int.= (0,-5) (0, 2) TEST (0, 4) (0-6) TEST (0,-4) 2 > ¼ (0)+3-3 > ¼ (0) < 3(0) < 3(0) > 3 4 > 3-6 < -5-4 < -5 FALSE TRUE TRUE FALSE Step 3: Label the area where the shading intersects with an S 51
52 2) y - 3 < - ⅓(x 6) 12x 6y > -12 3) x > 4 y < -5 52
53 4) 24x + 6x > -6 y > 2 5) y 6 < ⅔(x - 9) x < -3 53
54 Q3 Quiz 6 Review 1) y > 5 3x - y > -3 2) y + 6 > -½ (x - 8) y - 4 > 2(x 2) 54
55 3) 15x 45y < 90 x > 3 4) 21x 7y > 14 y- 3 > -¼ (x + 12) 55
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