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Sect 3.1 Reading Graphs How much money will you earn in a lifetime with an associate’s degree?$1.5 to 1.6 million What degree must you obtain to earn at least 2 million dollars in a lifetime? Bachelor’s or higher. Sect 3.1 Reading Graphs % is 100 of 9 x 100 134.8 Find the amount of 134.8 billion that was given to Pell Grants. x 9 134.8 100 x 12.132 billion Avg. 12,132,000,000 5,387,000 Avg. $2,252.09 Sect 3.1 Reading Graphs Indicates no visual comparisons! How many months of regular exercise are required to lower the pulse rate as much as possible? 6 MONTHS How many months of regular exercise are needed to achieve a pulse rate of 65 beats per min.? 3 MONTHS Sect 3.1 Reading Graphs All points are labeled (x, y). y P ( x, y ) A(__, 4 __) 3 B(__, 3 __) 5 C (__, 4 __) 3 D (__, __) 4 2 E (__, 1 __) 5 F (__, 2 __) 0 G (__, 3 0 __) x Sect 3.1 Reading Graphs These points are not in a Quadrant! Sect 3.2 Graphing Linear Equations Determine if (2, 5) and (-1, -1) are solutions to y 2x 1 2,5 y 2x 1 5 22 1 (2, 5) 5 4 1 55 TRUE Yes (2, 5) is a solution. 1,1 y 2x 1 1 21 1 1 2 1 1 1 TRUE Yes (-1, -1) is a solution. (-1, -1) Sect 3.2 Graphing Linear Equations When the equation is solved for y, y is the dependent variable and x is the independent variable. This means we can pick values for x and substitute them in to find the y value. Graph y 3x 2 Start with x = 0. Easiest to multiply by. x y 3x 2 (x, y) 0 y 30 2 y2 0,2 1 y 31 2 y 1 1,1 2 y 32 2 2,4 y 4 Use a straight edge to connect the points to form a straight line. Sect 3.2 Graphing Linear Equations Graph y 2 x 3 Start with x = 0. Easiest to multiply by. x y 2x 3 (x, y) 0 y 20 3 y 3 0,3 1 y 21 3 y 1 1,1 2 y 22 3 y 1 2,1 Use a straight edge to connect the points to form a straight line. Sect 3.2 Graphing Linear Equations 2 x 1 3 Graph y Start with x = 0, Count by 3’s due to the denominator in the fraction. x y 2 x 1 3 0 y 2 3 (x, y) 0 1 0,1 y 1 3 3 1 +3 +2 3,1 y 2 1 y 2 3 y 1 6 6 1 +3 +2 6,3 y 4 1 y 2 3 y3 Notice a pattern in the points? Y-coordinates and X-coordinates. The fraction contains the changes in the y-coord. and x-coord.! Sect 3.2 Graphing Linear Equations 1 Graph y x 3 2 Start with x = 0, Count by 2’s due to the denominator in the fraction. x 1 y x3 2 (x, y) 0 y 12 0 3 0,3 y 3 2 y 12 2 3 y 1 3 +2 -1 2,2 y2 4 y 12 4 3 y 2 3 +2 -1 4,1 y 1 Use a straight edge to connect the points to form a straight line. Sect 3.2 Graphing Linear Equations Graph 2 x 5 y 5 + 5 + 5y Solve the equation for y! + 5 + 5y 2x 5 5 y 5 5 5 2 x 1 y 5 x y 2 x 1 5 0 y 2 5 0 1 y 1 5 5 y 2 5 5 1 y 2 1 y3 y 52 5 1 y 2 1 y 1 (x, y) 0,1 5,3 5,1 Sect 3.3 Graphing Linear Equations in Standard Form Ax By C ; m Graph 2 x 5 y 5 m x-intercept ( 5 ____, 0 2 2x 50 5 ) 2x 5 x 5 2 1 ) y-intercept ( 0, ____ 20 5 y 5 5 y 5 y 1 2 2 5 5 A B Sect 3.3 Graphing Linear Equations in Standard Form Graph 2 x 4 y 12 2 1 m 4 2 6 0) x-intercept ( ____, 2x 40 12 2x 12 x6 y-intercept ( 0, ____ 3 ) 20 4 y 12 4 y 12 y 3 Sect 3.3 Graphing Linear Equations in Standard Form Graph 3 x 2 y 12 3 3 m 2 2 4 0) x-intercept ( ____, 3x 20 12 3x 12 x4 y-intercept ( 0, ____ 6 ) 30 2 y 12 2 y 12 y 6 Sect 3.3 Graphing Linear Equations in Standard Form Graph 4 x 5 y 0 4 4 m 5 5 0 0) x-intercept ( ____, 4 x 50 0 4x 0 x0 y-intercept ( 0, ____ 0 ) 40 5 y 0 5y 0 y0 Sect 3.3 Graphing Linear Equations in Standard Form Rewrite in standard form: Ax + By = C Graph 4 x 8 8 y 4 x 8 y 8 Short Cut: Cover-up technique. 2 0 ) x-intercept ( ____, To find the x-intercept, cover-up the y term. Solve for x. x 2 1 ) y-intercept ( 0, ____ To find the y-intercept, cover-up the x term. Solve for y. y 1 m 4 1 8 2 Sect 3.3 Graphing Linear Equations Notice there is no y variable in the equation…the line can’t cross the y-axis. No y-intercept. 2x 3 11 2x 8 x4 4 0) x-intercept ( ____, Graph y-intercept ( 0, ____ ) Sect 3.3 Graphing Linear Equations Notice there is no x variable in the equation…the line can’t cross the x-axis. No x-intercept. Graph 3y 6 y2 x-intercept ( ____, 0 ) 2 ) y-intercept ( 0, ____ Sect 3.3 Graphing Linear Equations Graph 5 x 3 y 11 11 x-intercept ( ____, 0) 5 YUCK! 11 y-intercept ( 0, ____ 3 ) Lets find clean points. Take the LARGEST coefficient and take it’s opposite to the other side. 5 x 3 y 11 5x 5x 3 y 11 5 x Start with 11 and keep subtracting by 5 until you have a number divisible by 3. 11 5 6 We only subtracted by one 5, so x = 1. 3 y 6, when x 1 y2 A 5 1, 2 m B 3 Sect 3.3 Graphing Linear Equations Graph 7 x 5 y 19 19 x-intercept ( ____, 0) 7 19 y-intercept ( 0, ____ 5 ) YUCK! Lets find clean points. Take the LARGEST coefficient and take it’s opposite to the other side. 7 x 5 y 19 5 y 19 7 x 7 x 7 x Start with 19 and keep subtracting by 5 until you have a number divisible by 5. 19 7 12 We subtracted by two 7’s, so x = 2. 12 7 5 5 y 5, when x 2 y 1 2, 1 m A 7 7 B 5 5 Sect 3.4 Rates A Rate is a ratio that indicates how two quantities change with respect to each other. Unit rate is when the second quantity is one. On Jan. 3rd, Joe rented a car with a full tank of gas and 9312 miles on the odometer. On Jan. 7th, he returned the car with 9630 miles on the odometer. If Joe had to pay $108 for the total bill which included 12 gallons of gas, find the following rates. Convert to unit rates. 9630 9312 318miles Divide for unit rate. 318miles miles 26.5 gallons 12 gallons $108 $21.6 2. Average cost of the rental in dollars per day. 5days day rd th th th th 1. Gas consumption in miles per gallon. Jan. 3 , 4 , 5 , 6 , and 7 = 5 days 3. Travel rate in miles per day. 318miles 5days 63.6miles day Sect 3.5 Slope The Slope is a ratio that indicates how the change in the y-coordinates change with the respect to the change in the x-coordinates. The slope of a line contains two points ( x1, y1 ) and ( x2, y2 ) is given by rise y2 y1 m change in x run x2 x1 change in y Change in x Change in y x1 , y1 x2 , y2 Sect 3.5 Graphing Linear Equations Graph ( -4, 3 ) and ( 2, -6 ) and find the slope. m 9 3 6 2 rise y2 y1 m run x2 x1 63 2 4 9 3 m 6 2 m Change in y = -9 Change in x = 6 Sect 3.5 Graphing Linear Equations m3 m2 Positive slopes always have lines that go uphill. Slopes > 1 are steep. 0 < Slopes < 1 begin to flatten out. m 1 m 1 2 1 m 4 m 2 3 Sect 3.5 Graphing Linear Equations m 1 m 2 m 3 Negative slopes always have lines that go downhill. Slopes < -1 are steep. -1 < Slopes < 0 begin to flatten out. 1 m 4 1 m 2 2 m 3 Sect 3.5 Graphing Linear Equations Graph ( -4, 3 ) and ( 2, 3 ) and find the slope. 33 2 4 0 m 0 6 Horizontal Line m Graph ( 5, -1 ) and ( 5, 6 ) and find the slope. Vertical Line 6 1 55 7 m undefined 0 m Sect 3.5 Slope The Grade is a slope that is measured as a percent. 7% 7 vertical ft 100 horizontal ft Drop 7 feet for every 100 feet traveled horizontally. Sect 3.6 Graphing Linear Equations in Slope Intercept Form Graph y 2 x4 3 ( 0, - 4 ) Starting point Directions 2 up & 3 right Or opposite 2 down & 3 left Sect 3.6 Graphing Linear Equations in Slope Intercept Form Graph 5 y x6 2 ( 0, 6 ) Starting point Directions 5 down & 2 right 2x y 3 -3 + y = -3 + y 2x 3 y y 2x 3 1 ( 0, -3 ) Starting point Directions 2 up & 1 right Opposite 2 down & 1 left Sect 3.6 Graphing Linear Equations in Slope Intercept Form Solving for y would not be a good decision because it will generate a fraction for a y-intercept. Graph 5 x 3 y 10 x-intercept and slope. 2,0 A 5 5 m B 3 3 5 x 4 y 11 x-intercept and y-intercept are bad…fractions. 4 y 11 5 x 11 5 6 6 5 1 1 5 4 x = 3 for the three 5’s we subtracted. 4 y 4 y 1; 3,1 m A 5 5 B 4 4 Sect 3.6 Graphing lines Parallel Lines – Lines are parallel when they have the same slopes or the lines are vertical. m1 m2 Perpendicular Lines – Lines are perpendicular when their slopes are opposite reciprocals of each other. m1 m2 1 I prefer to say, “the perpendicular slopes are opposite reciprocals.” Sect 3.7 Graphing Linear Equations in Point Slope Form Consider a line with slope 2 passing through the point ( 4, 1 ). y 1 m 2 x4 Consider removing the “m” and re-writing so there is no fraction. y 1 2 x4 y 1 2 x 4 x 4 x4 ( x, y ) ( 4, 1 ) y 1 2 x 4 Notice the sign change! The equation y y1 mx x1 is called point-slope form of a linear equation with the slope m and the point x1 , y1 . Sect 3.7 Graphing Linear Equations in Point Slope Form Graph 3 y 5 x 1 4 Start @ (1, 5) m = -3 4 y2 Start @ (-4, 2) 1 x 4 3 1 m = 3 3 y 5 x 4 2 3 Start @ (-4, -5) m = 2 y 5 4x 2 Start @ (2, -5) m = 4 1 Sect 3.7 Graphing lines y y1 mx x1 5 Write the equation of a line in point-slope form with the slope of and the 7 point ( -14, 12). Convert to slope-intercept form. 5 5 y 12 x 14 y 12 x 10 7 + 12 = 7 + 12 5 5 2 y 12 x 14 7 7 5 y x 22 7 8 Write the equation of a line in point-slope form with the slope of and the 5 point ( 4, -6). Convert to slope-intercept form. 8 x 4 y 6 8 x 32 5 5 5 –6 = –6 8 8 y 6 x 4 5 5 y6 8 32 30 y x 5 5 5 8 2 y x 5 5 Sect 3.7 Graphing lines…ANOTHER APPROACH 8 Write the equation of a line in point-slope form with the slope of and the 5 point ( 4, -6). Convert to slope-intercept form. NO FRACTION WORK…Remember Standard Form. A 8 m B 5 A8 B5 Ax By C ; m 8x 5 y C Substitute the point ( 4, -6 ) for x and y to solve for C. 84 5 6 C 32 30 C 2C 8x 5 y 2 – 8x = – 8x 5 y 8 x 2 5 5 Short cut. b 8x 5 y C Solve for y. A B 5 C 2 B 5 8 2 y x 5 5 Sect 3.7 Graphing lines Ax By C ; m Same slopes. A B Write the equation of a line that is parallel to 3x + 5y = 11 and contains the point ( 0, 2). Use slope-intercept form. A 3 y-intercept (0, b) y mx b m B 5 3 y x2 5 Opposite and Reciprocal slopes. Write the equation of a line that is perpendicular to 3x + 5y = 11 and contains the point ( 0, 2). Use slope-intercept form. A 3 m y-intercept (0, b) B Perpendicular slope. 5 y x2 3 B 5 m A 3 5 Sect 3.7 Graphing lines Must be the same!. Same slopes. Write the equation of a line that is parallel to 7x – 4y = 9 and contains the point A 7 C 47 ( 5, -3). Use slope-intercept form. Ax By C ; m 7x 4 y C 75 4 3 C 35 12 C 47 7 x 4 y 47 4 y 7 x 47 4 4 4 7 47 y x 4 4 B y 4 b B 4 7 47 x 4 4 Write the equation of a line that is perpendicular to 8x + 3y = 13 and contains the point ( -7, 4). Use slope-intercept form. A 8 Perpendicular slope. m The numbers must flip and change the sign! 8 x 3 y 13 B 3 3 x 8 y C New. 3x 8 y 53 3 7 84 C 21 32 C 53 3 53 y x 8 8 B 3 A 8 C 53 53 b B 8 8 m Sect 9.4 Graphing Linear Inequalities 3 Graph the line y x 3 . 2 Consider changing the equation to an inequality. 3 y x3 2 Determine if the points are solutions, ( -4, 3 ), (-8, 5 ), and ( 2, -1 ). 3 3 5 8 3 1 2 3 2 2 5 12 3 1 3 3 59 1 6 FALSE TRUE Sect 9.4 Graphing Linear Inequalities in Slope Intercept Form Equal to line means a solid line! No equal to line means a dashed line! Graph 5 y x6 2 The inequality has y > which tells me to shade all the y coordinates greater than the line…above the line. Solve for y. 2x y 3 – 2x = – 2x y 2 x 3 -1 -1 -1 y 2x 3 Remember to flip the inequality symbol. Shade all the y coordinates less than the line…below the line. Sect 9.4 Graphing Linear Equations Graph the line 2 x 4 y 8 . Find the x and y intercepts. Consider changing the equation to an inequality. 2 x 4 y 8 Determine which way to shade. Test the Origin (0, 0) 20 40 8 0 8 TRUE Shade in the direction of the Origin Sect 9.4 Graphing Linear Equations in Standard Form Graph 4 x 3 y 24 Find the x and y intercepts. Test the Origin (0, 0) 40 30 24 0 24 TRUE 2 x 4 y 16 Find the x and y intercepts. Test the Origin (0, 0) 20 40 16 0 16 FALSE Sect 9.4 Graphing Linear Equations Graph 6 y 24 y4 2 x 12 x 6 Sect 9.4 Graphing Linear Equations Graph 3 y 2