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Chapter 10 Graphing Equations and Inequalities 1 Section 10.1 Rectangular Coordinates 2 Rectangular Coordinate System Also called Cartesian, after René Descartes y x Ordered Pair (3, 4); the point where x = 3 and y = 4 3 Example • Plot the following: (-2, 4), (-1, -1), (3, 4), (1, -3) 4 Solution • Plot the following: (-2, 4), (-1, -1), (3, 4), (1, -3) 5 Identify the Points 1 1 6 Solution • (4,3) • (-3, 5) • (-4, -3) • (-1, -4) • (3, - 4) 7 Draw a Scatter Diagram Year 2003 2004 2005 2006 2007 2008 2009 2010 Sales, $M 3 3.4 3.8 4.2 4.6 5 5.4 5.8 8 Solution Sales, $M 3 3.4 3.8 4.2 4.6 5 5.4 5.8 7 6 5 Sales, $M Year 2003 2004 2005 2006 2007 2008 2009 2010 4 3 2 1 0 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 Year 9 Draw a Scatter Graph Year 2003 2004 2005 2006 2007 2008 2009 Tornadoes 1376 1817 1264 1106 1098 1691 1156 10 Solution Year 2003 2004 2005 2006 2007 2008 2009 Tornadoes 1376 1817 1264 1106 1098 1691 1156 2000 Number of Tornadoes 1800 1600 1400 1200 1000 800 2002 2003 2004 2005 2006 Year 2007 2008 2009 2010 11 Why Make a Graph? • Spot trends – Are your grades improving? – How is the stock market doing? – Are we really experiencing global warming? • Can compare two lines and find differences • Allows us to transfer information quickly – we understand pictures better than we understand lists of numbers 12 Standard Form of Grid; Terminology 13 Give the quadrant of the following coordinates • (3, 4) • (-2, 4) • (2, -5) • (3, -5) • (-1, -1) 14 Solution • (3, 4) Q1 • (-2, 4) Q2 • (2, -5) Q4 • (3, -5) Q1 • (-1, -1) Q3 15 Ordered Pairs are Solutions to an Equation • 2x + y = 8 or y = 8 – 2x – The coordinate pair (0, 8) or x = 0, y = 8 is a solution – The coordinate pair (1, 8) or x = 1, y = 8 is not a solution 16 Example • y = 12 – 3x x -2 -1 0 1 2 3 y = 12 - 3x 17 Solution • y = 12 – 3x x -2 -1 0 1 2 3 y = 12 - 3x 18 15 12 9 6 3 18 Example • y = 3x x -2 -1 0 1 2 3 y = 3x 19 Solution • y = 3x x -2 -1 0 1 2 3 y = 3x -6 -3 0 3 6 9 20 Example • y = (1/2) x - 5 x -2 -1 0 1 2 3 y = 1/2 x - 5 21 Solution • y = (1/2) x - 5 x -2 -1 0 1 2 3 y = 1/2 x - 5 -6 -5.5 -5 -4.5 -4 -3.5 22 Section 10.2 Graphing Linear Equations 23 Linear Equations • ay = bx + c • a, b and c are any numbers • A graph of a linear equation in two variables results in a straight line when graphed in a coordinate plane • Examples: y = 4x – 3 9y = -4x + ½ y/2 = -3/8 x + 5000 • The terms can be on different sides of the equal sign: x = 5y + 2; 3 = x + y 24 Are these linear equations? • y = 3x – 4 • x=4 • y = 2(x - 3) + 2 • x = 5y – 3(y + 2) • y = 3x2 – x + 5 • x = y + 21 25 Graphing • Make a table of at least three values • Plot points • Join with line 26 Example • y = 12 – 3x 27 Solution y = 12 - 3x 18 15 12 9 6 3 20 16 12 8 y x -2 -1 0 1 2 3 4 0 -4 -2 -4 0 2 4 x 28 • y = 3x 29 y = 3x 4 3 y -3 0 3 2 1 0 y x -1 0 1 -2 -1 -1 0 1 2 -2 -3 -4 x 30 • Y = 1/2 x - 5 31 y = 1/2 x - 5 -6 -5.5 -5 -4.5 -4 -3.5 2 0 -4 y x -2 -1 0 1 2 3 -2 0 2 4 -2 -4 -6 x 32 • y = -2 33 • y = -2 x y -1 0 1 2 -2 -2 -2 -2 1 -2 -1 0 1 2 3 -1 -3 34 Section 6.3 Intercepts and Special Lines 35 Intercepts • An intercept is the value where a line crosses an axis x intercept is the value where a line crosses the x axis y intercept is the value where a line crosses the y axis • We write these values as coordinate pairs: x intercept is (x, 0), since the x axis is at y = 0 y intercept is (0,y), since the y axis is at x = 0 36 Find the x Intercept • 3x – y = 4 • 2x + 5y + 1 = 0 • x=¾y+5 37 Solution x intercept • 3x – y = 4 y=0 x = 4/3 (4/3, 0) • 2x + 5y + 1 = 0 2x = -1 – 5y y=0 x = -1/2 (-1/2, 0) • x=¾y+5 y=0 x=5 (5, 0) 38 Find the y intercept • 3x – y = 4 • 2x + 5y + 1 = 0 • x=¾y+5 39 Solution • 3x – y = 4 x=0 y = -4 (0, -4) • 2x + 5y + 1 = 0 x=0 5y + 1 = 0, y = -1/5 (0, -1/5) • x=¾y+5 x=0 ¾ y = -5, y = -20/3 (0, -20/3) 40 Using the Intercept to Graph • 2x + 5y + 1 = 0 • Intercepts are: (-1/2, 0) and (0, -1/5) • These can be two of your three points 41 Graph Using Intercepts • y=x–3 Intercepts are (0, -3) and (3, 0) • Use one more point as a check: y = 1, x = 4 42 Graph Using Intercepts • 3x – 4y = 12 43 Solution • 3x – 4y = 12 Intercepts are (0, -3), (4, 0) One additional point: x = 2, y = -3/2 44 Vertical Lines • x = 5 is a vertical line through x = 5 it has no y intercept 45 Horizontal Lines • y = 4 is a horizontal line through y = 4 it has no x intercept 46 Are the Following Lines Horizontal, Vertical, or Neither? • 3x = 4x + 2 • y = 4-3x • 4 + 4y = y + 3 • 2y + x = 2y + 4 • x=x 47 Solutions • 3x = 4x + 2 Vertical • y = 4-3x Line, neither vertical or horizontal • 4 + 4y = y + 3 Horizontal Line • 2y + x = 2y + 4 Vertical line (x drops out) • x = x Not a line at all 48 Chapter 10.4 Slope and Rate of Change Background • We can often relate the behavior of two quantities by a line – The amount you eat to the amount you weigh – The amount of sun to the temperature – The gas setting on the stove to how fast the kettle boils • Today we address the relative change of the two quantities we are examining An Example 90 80 60 50 40 30 90 20 80 10 70 2008 2009 Year A 2010 2011 60 50 40 160 30 140 20 120 10 0 2006 2007 2008 2009 2010 Year 2011 Stock Price 0 2006 2007 Stock Price Stock Price 70 100 80 60 40 B Which Company’s Stock is Growing Fastest? 20 0 2006 2007 2008 2009 2010 2011 Year C 51 An Example 90 80 60 50 40 30 90 20 80 10 70 2008 2009 Year A 2010 2011 60 50 40 160 30 140 20 120 10 0 2006 2007 2008 2009 2010 Year 2011 Stock Price 0 2006 2007 Stock Price Stock Price 70 100 80 60 40 B Hint: Examine the axes; They are not the same 20 0 2006 2007 2008 2009 2010 2011 Year C 52 How can we compare growth? • In this case, the growth is illustrated by comparing the steepness of the line • When trying to compare the steepness of a line, we look at a quantity we define as slope Slope m = change in y / change in x = vertical change / horizontal change Could we do it the opposite? Yes, but we are used to thinking of things going up and down. 53 Finding Slope • Pick two points (x1, y1) and (x2, y2) • Find 𝑦1−𝑦2 𝑥1−𝑥2 NOTE: You have to keep the order, 1 and 2, the same in the numerator and the denominator!! 160 90 140 80 120 Stock Price Stock Price 70 60 50 40 30 80 60 40 20 20 10 0 2006 100 2007 2008 2009 2010 2011 0 2006 2007 2008 2009 2010 2011 Year Year B C 54 Sample Points • In company B, point 1 is (2007, 20), point 2 is (2009, 80) • In company C, point 1 is (2007, 20), point 2 is (2009, 140) Company B: Slope = 60/2 = 30 Company C: Slope = 120/2 = 60 – far larger slope 160 90 140 80 120 Stock Price Stock Price 70 60 50 40 30 80 60 40 20 20 10 0 2006 100 2007 2008 2009 2010 2011 0 2006 2007 2008 2009 2010 2011 Year Year B C 55 Slope • The ratio of the change in y (height) to the change in x (distance) • We use the letter m to indicate slope 56 Example Find the slope of a line that passes through the following points: (-5, 8) and (3, -2) 57 Solution Find the slope of a line that passes through the following points: (-5, 8) and (3, -2) 8−(−2) −10 −5 = = −5−3 8 4 58 Does it matter which point is 1 and which is 2? Find the slope of a line that passes through the following points: (-5, 8) and (3, -2) 8−(−2) −5−3 −2−8 3−(−5) = −10 8 = −10 8 = −5 4 = −5 : 4 No difference! 59 Example Find the slope: Point 1: (-2, 0) Point 2: (0,5) 60 Solution Point 1: (-2, 0) Point 2: (0,5) m = 5/2 Does it matter which points we choose? • In the previous chart, we had coordinates (0, 5), (-2,0) and (-4, -5) • Let’s try to find the slope: From the first two: 5−0 = 0−(−2) From the last two: 0−(−5) = −2−(−4) From the first and last: 5/2 5/2 5−(−5) = 0−(−4) 10/4 = 5/2 • Any pair of points on a line give the same slope. What does the sign of a slope mean? • Positive slope: • Negative slope: 63 Sign of a slope • Positive slope: The line is going uphill • Negative slope: The line is going downhill 64 Slope from a Table x 0 5 10 15 20 25 30 y -4 -2 0 2 4 6 8 m =? 65 Can use any two points m =? x 0 5 10 15 20 25 30 y -4 -2 0 2 4 6 8 m = 2/5 66 Standard Form We can always put a linear equation into the form: y = mx + b, where m and b are numbers It turns out that if the equation is in this form, m is the slope! 67 Example of Slope • y = 4x – 8 • Find two points on the line: if y = 0, then x = 2; (2, 0) if x = 0, the y = -8; (0, -8) • Find slope: (0 – (-8))/ (2 – 0) = 8/2 = 4 • In the form y = mx + b = 4x =8, 4 is the slope More General Derivation y= mx + b Let two points on the line be (x1, y1) and (x2, y2) y1 = m(x1) + b y2 = m(x2) + b y1 – y2 = m (x1 – x2) + (b – b) (y1 – y2) = m (x1 – x2) (y1 – y2) / (x1 – x2) = m, the slope Example • -2x + 3y = 11 • Write in the form y = mx + b • -2x + 3y = 11 add 2x to each side: 3y = 2x + 11 divide both sides by 3: y = 2/3 x + 11/3 • Y = 2/3 x + 11/3, m = 2/3, the slope Find the slope • y = 5x -4 • 3y = -x + 5 • 2y = 4x + 8 71 Solution • y = 5x -4; m = 5 • 3y = -x + 5; m = -1/3 • 2y = 4x + 8 m = 2 72 Finding a Slope from a Line • Need to find the coordinates of two points on a line 4 3 2 1 y 0 -2 -1 -1 0 1 2 -2 -3 -4 x • (-1, -3) (0,0)(1, 3) 73 Finding Slope • (-1, -3) (0,0)(1, 3) • (y1 – y1) / (x1 – x2) = m • (0-(-3)) / (0-(-1)) = 3/1 = 3 Slope is 3. Can check with other points Horizontal Line Slope is Zero Choose any point: the y value is the same y1 – y2 = 0; 0/n = 0 Y = constant 75 Horizontal Line • Slope zero means m = 0 • e.g., y = 5 76 Vertical Line • Slope is 𝑦1−𝑦2 𝑥1−𝑥2 All values of x are the same; x1 – x2 = 0 The denominator is 0, so the fraction, and the slope are undefined 77 Vertical Line • x=5 • If we have a constant x, x cannot change. – Means x1 – x2 = 0 for all values of x – 𝑦1−𝑦2 , 𝑥1−𝑥2 x1-x2=0, so we have 𝑛 0 – Slope is undefined! 78 Recap • The slope of a line tells how steep it is – Larger slope is steeper – Negative slope is downhill • To find the slope, take two points on the line and calculate the difference in the y values, and divide by the difference in the x values: 𝑦1−𝑦2 𝑥1−𝑥2 = m, the slope – Be sure you keep the points in the right order! • Standard form is y = mx + b, where m is the slope 79 More Recap • A horizontal line, y= const, has slope zero – It doesn’t go up or down • A vertical line, x = const, has an infinite, or undefined slope – It goes up faster than we can calculate 80 Review Examples Find the slope: • 3x – 5y = 1 • x = -4y • -4x – 7y = 9 • x=1 • y = -3 Solutions • 3x – 5y = 1; m = 3/5 • x = -4y; m = -1/4 • -4x – 7y = 9; m = - 4/7 • x = 1; mis undefined • y = -3; m=0 Find the slope of the following lines • y = 5x – 3 • 3y = 4x • 2y – x = 4 – x • 3x = y -4 • 5y = x + 5y + 4 83 Solutions • y = 5x – 3 m=5 • 3y = 4x m = 4/3 • 2y – x = 4 – x 2 y = 4, horizontal line, slope zero: m = 0 • 3x = y -4 m=3 • 5y = x + 5y + 4 0 = x + 4, x = -4, vertical line, undefined slope, m undefined 84 Deriving a Linear Equation 85 Linear Equations • Standard form: y = mx + b m is the slope b is the y intercept, where the line intersects the x axis x=0 • Can find the equation for a line given A. A Slope and a Point B. Two points (from two points can find a slope, then do A) 86 Example • Find the line with slope 2 that contains the point (4, 6) 87 Solution • Find the line with slope 2 that contains the point (4, 6) • y = mx + b • Substitute for m: y = 2x + b • Put in 4 and 6: 6 = 2(4) + b = 8 + b b = -2 since 6 = 8 – 2 • y = 2x - 2 88 Example • Find the line with slope = -1/2 that passes through the point (0, 4) 89 Solution • Find the line with slope = -1/2 that passes through the point (0, 4) • y = mx + b; substitute for m y = -x/2 + b • Put in the point (0, 4) 4 = -(0)/2 + b = 0 + b b=4 • y = -x/2 + 4 • Can check your answer: 4 = -0/2 + 4 90 Example • Find the equation of line that contains the points (3, 5) and (6, -1) 91 Solution • Find the equation of line that contains the points (3, 5) and (6, -1) • m = (y1 – y2) / (x1 – x2) = (5 – (-1))/(3 – 6)= 6/(-3) = -2 • y = -2x + b put in either of the points given • 5 = -2(3) + b = -6 + b b = 11 • y = -2x + 11 • Check solution: -1 = -2(6) + 11 = -12 + 11 = -1 5 = -2(3) + 11 = -6 + 11 = 5 92 Example • Find the equation of the line that contains the points (4, -5) and (-3, 1) 93 Solution • Find the equation of the line that contains the points (4, -5) and (-3, 1) • m = (-5 – 1)/(4 – (-3)) = -6/7 • y = -6/7 x + b • Plug in points: 1 = (-6/7)(-3) + b = -18/7 + b 1 - 18/7 = 7/7 - 18/7 = -11/7 = b • y = -6/7x - 11/7 • Check: -5 = (-6/7)(4) - 11/7 = -24/7 - 11/7 = -35/7= -5 1 = (-6/7)(-3) – 11/7 = 18/7 – 11/7 = 7/7 = 1 94 Examples • (0, 4) and (5, -1) are on a line: find the equation of the line 95 Solution • (0, 4) and (5, -1) are on a line: find the equation of the line • Find slope: 96 Some More Special Lines 97 Parallel Lines • Parallel lines have the same slope, yet different intercepts • 5y = 3x – 4 • 5y = 3x – 8 • Let’s graph these lines 98 2 1 y 0 -5 -4 -3 -2 -1 -1 0 1 2 3 4 5 -2 -3 -4 -5 x 5y = 3x - 4 5y = 3x - 8 Perpendicular Lines • Perpendicular lines have slopes that are negative reciprocals • Line 1 has slope 3, Line 2 has slope – 1/3. These are perpendicular lines • 3x = y + 5 • x = -3y + 10 100 Perpendicular Example • 3x = y + 5 m=3 • x = -3y + 10 m = - 1/3 (-1/3 ) ( 3) = 1 101 Graph these lines using slope • 3x = y + 5 • x = -3y + 10 Solution • 3x = y + 5; same as y = 3x - 5 • x = -3y + 10; same as y = -x/3 + 10/3 8 4 x = -3y + 10 3x = y + 5 y 0 -8 -4 0 -4 -8 x 4 8 Are these equations parallel, perpendicular, or neither • y = 4x – 2 4x + y = 5 • 6 + 4x = 3y 3x + 4y = 8 • 6x = 5y + 1 -12x + 10y = 1 Solution • y = 4x – 2 4x + y – 5 neither • 6 + 4x = 3y 3x + 4y = 8 perpendicular • 6x = 5y + 1 -12x + 10y = 1 parallel Review: Solve each equation for y • y – 7 = -9 ( x – 6 ) • y – ( - 3) = 4 (x – (- 5)) Section 10.5 Equations of Lines Using the Intercept to Graph • 2x + 5y + 1 = 0 • Intercepts are: (-1/2, 0) and (0, -1/5) • These can be two of your three points 108 Graph Using Intercepts • y=x–3 Intercepts are (0, -3) and (3, 0) • Use one more point as a check: y = 1, x = 4 109 Graph Using Intercepts • 3x – 4y = 12 110 Solution • 3x – 4y = 12 Intercepts are (0, -3), (4, 0) One additional point: x = 2, y = -3/2 111 Graphing Given a Point and a Slope • Find the point • Count up and over, depending on the slope • For example: graph the line with slope 3 that contains (1,2) 1. Plot the point (1,2) 2. Count up 3 over 1 from that point to (2, 5) NOTE: If the slope is a whole number, the denominator is 1 Example • Graph the line that passes through (-2, 3) with slope 5 113 Solution Graph the line that passes through (-2, 3) with slope 5 1. Plot the point (-2, 3) 2. Count up 5 and over 1 to (-1, 8) and connect the points 114 Example • Graph a line with slope 1/3 that passes through (-1, -2) 115 Solution • Graph a line with slope 1/3 that passes through (-1, -2) 1. Plot the point (-1, -2) 2. From there go up 3 and over 1 to get to (1, 1) 3. Connect the points 116 Example • Graph the line of slope – 4/3 that passes through (-1, 8) 117 Solution • Graph the line of slope – 4/3 that passes through (-1, 8) 1. Plot the point (-1, 8) 2. From that point go down 4 and over 3 to get to (2, 4) 3. Connect the points • Note: we could have also gone up 4 and to the left three from our point and reached (-5, 12). This point is also on the line 118 Graph from a Slope and an Intercept • 4x + y = 1 • First put in format y = mx + b y = -4x + 1 • y Intercept is (0,1) • Slope, m is -4 • Find intercept, and count down 4 and over 1 Example • 4x + y = 1 • y= - 4x+1 intercept is (0, 1), slope is – 4 120 What you need to know how to do: • Given the points, find the slope and write the equation • Given the slope and a point, write the equation • Given the slope and the y intercept, write the equation • And, how to graph these equations