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Slide 2 Point Slope Math 374 Slide 3 Topic Box 1) Slope 1) Slope 2) Word Problems 2) Word Problems - keys - Point Point - Slope Point - Y Int Point - Slope Y int Slide 4 The Slope We know that we are dealing with linear functions We know that we are dealing with linear functions y = mx + b y = mx + b Slope y int The slope is not always obvious, if we do not know the equation Slide 5 Understanding the Slope What makes a slope? What makes a slope? Rise Run We define the slope as the ratio between the rise and the run Slope = m = rise run Slide 6 Formula for Slope If we have two points (x 1, y 1 ) (x 2, y 2 ) If we have two points (x 1, y 1 ) (x 2, y 2 ) Slope = m = y 1 y 2 = y 2 y 1 Slope = m = y 1 y 2 = y 2 y 1 x 1 x 2 x 2 x 1 Remember it is Y over X! Remember it is Y over X! Maintain order Maintain order Slide 7 Calculating Slope Slope = m = y 1 y 2 = y 2 y 1 Slope = m = y 1 y 2 = y 2 y 1 x 1 x 2 x 2 x 1 x 1 x 2 x 2 x 1 (5, 4) (2, 1) (5, 4) (2, 1) 4 - 1 4 - 1 5 - 2 5 - 2 3 3 m = 1 m = 1 (x 1,y 1 ) (x 2,y 2 ) Slide 8 Ex # 2 A = (-4, 2) B=(2, -4) (x 1,y 1 ) (x 2,y 2 ) -4 2 2 - - 4 -6-6 6 m = -1 Slide 9 (4, 5) (1, 1) Ex #3 (x 2,y 2 ) (x 1,y 1 ) 1 5 1 4 m = -4 -3 Slide 10 Two Decimal Place Practice Find the slope of (4,217.08) (11,575.41) Find the slope of (4,217.08) (11,575.41) m = 217.08 575.41 m = 217.08 575.41 4-11 4-11 51.19 51.19 Slide 11 Word Problems BASICS Set your x and y Set your x and y Find the linear function meaning find slope m and y int or b Find the linear function meaning find slope m and y int or b Solve question Solve question x is always time x is always time y is always money y is always money Slide 12 Word Problems Cont KEYS Per Per Rate Rate m Fixed Initial b Point two values point Slide 13 Point Point Little Billy rents a car for 5 days and pays $463.65. Little Sally rents a car for 14 days and pays $1208.22 Little Billy rents a car for 5 days and pays $463.65. Little Sally rents a car for 14 days and pays $1208.22 a) How much would it cost to rent a car for 27 days? b) How many days can you get for $2531.90 Step 1: Set Your x and y what are they? Slide 14 Solution x is # of days x is # of days y is $ (money) y is $ (money) (5, 463.65) (14, 1208.22) (5, 463.65) (14, 1208.22) m = 463.65 1208.22 m = 463.65 1208.22 5 14 5 14 m = 82.73 m = 82.73 y = 82.73x + b y = 82.73x + b Slide 15 Solution Cont (5, 463.65) 463.65 = 82.73(5) + b (5, 463.65) 463.65 = 82.73(5) + b 463.65 = 413.65 + b 463.65 = 413.65 + b 50 = b 50 = b b = 50 b = 50 y = 82.73x + 50 y = 82.73x + 50 A) x = 27 y = 82.73 (27) + 50 A) x = 27 y = 82.73 (27) + 50 y = $2283.71 y = $2283.71 Slide 16 Solution Cont b) y = 2531.90 2531.90 = 82.73x + 50 b) y = 2531.90 2531.90 = 82.73x + 50 2481.90 = 82.73x 2481.90 = 82.73x 30 = x 30 = x x = 30 x = 30 Slide 17 Slope Point Little Billy rents a car and pays a rate of $43.91 per day. Little Sally rents for 42 days and pays $1900.45 Little Billy rents a car and pays a rate of $43.91 per day. Little Sally rents for 42 days and pays $1900.45 a) How much would it cost to rent a car for 27 days? a) How much would it cost to rent a car for 27 days? b) How many days can you get for $2471.28? b) How many days can you get for $2471.28? Step 1: Set your x and y Step 1: Set your x and y Slide 18 Solution x is # of days x is # of days y is $ (money) y is $ (money) m = $43.91 (42,1900.45) m = $43.91 (42,1900.45) y = 43.91x + b y = 43.91x + b (42,1900.45) 1900.45 = 43.91 (42) + b (42,1900.45) 1900.45 = 43.91 (42) + b 1900.45 = 1844.22 + b 1900.45 = 1844.22 + b 56.23 = b 56.23 = b b = 56.23 b = 56.23 Slide 19 Solution Cont y = 43.91x + 56.23 y = 43.91x + 56.23 A) x = 27 y = 43.91 (27) + 56.23 A) x = 27 y = 43.91 (27) + 56.23 y = $1241.80 y = $1241.80 B) y = 2471.28 2471.28=4391 x+56.23 B) y = 2471.28 2471.28=4391 x+56.23 2415.05 = 43.91x 2415.05 = 43.91x 55 = x 55 = x x = 55 x = 55 Slide 20 Y Int - Point Little Billy rents a car and pays a fixed cost of $132.11. Little Sally rents for 12 days and pays $1331.99 Little Billy rents a car and pays a fixed cost of $132.11. Little Sally rents for 12 days and pays $1331.99 a) How much for 27 days? a) How much for 27 days? b) How many days for $1232? b) How many days for $1232? x is # of days x is # of days y is $ money y is $ money b = 132.11 (12, 1331.99) b = 132.11 (12, 1331.99) y = mx + 132.11 y = mx + 132.11 Slide 21 Solution (12, 1331.99) 1331.99 =12m+132.11 (12, 1331.99) 1331.99 =12m+132.11 1199.88 = 12m 1199.88 = 12m 99.99 = m 99.99 = m m = 99.99 m = 99.99 y = 99.99x + 132.11 y = 99.99x + 132.11 a) x = 27 y = 99.99 (27) + 132.11 a) x = 27 y = 99.99 (27) + 132.11 y = $2831.84 y = $2831.84 Slide 22 Solution (Cont) B) y = 1232 1232 = 99.99x + 132.11 B) y = 1232 1232 = 99.99x + 132.11 1099.89 = 99.99x 1099.89 = 99.99x 11 = x 11 = x x = 11 x = 11 Slide 23 Slope Y int Little Billy rents a car and pays a rate of $73.99 per day. Little Sally pays a fixed cost of $42.19 Little Billy rents a car and pays a rate of $73.99 per day. Little Sally pays a fixed cost of $42.19 a) How much for 27 days? a) How much for 27 days? b) How many days for $412.14 b) How many days for $412.14 x is # of days x is # of days y is $ (money) y is $ (money) m = 73.99 b = 42.99 m = 73.99 b = 42.99 y = 73.99x + 42.19 y = 73.99x + 42.19 Slide 24 Solutions x= 27 y = 73.99 (27) + 42.19 x= 27 y = 73.99 (27) + 42.19 y = 2039.92 y = 2039.92 b) y = 412.14 412.14 = 73.99x+42.19 b) y = 412.14 412.14 = 73.99x+42.19 369.95 = 73.99x 369.95 = 73.99x 5 = x 5 = x x = 5 x = 5