point slope
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Point SlopePoint Slope
Math 374Math 374
Topic BoxTopic Box
• 1) Slope1) Slope
• 2) Word Problems2) Word Problems
- keys- keys
- Point – Point - Point – Point
- Slope – Point- Slope – Point
- Y Int – Point- Y Int – Point
- Slope – Y int- Slope – Y int
The SlopeThe Slope
• We know that we are dealing with We know that we are dealing with linear functionslinear functions
y = mx + by = mx + b
Slope
y int
The slope is not always obvious, if we do not know the equation
Understanding the SlopeUnderstanding 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
Formula for SlopeFormula for Slope
• If we have two points (xIf we have two points (x11, y, y11) (x) (x22, y, y22))
• Slope = Slope = mm = = yy11 – y – y22 = = yy22 – y – y11
xx11 – x – x2 2 x x2 2 – x – x11
• Remember it is Y over X! Remember it is Y over X!
• Maintain orderMaintain order
Calculating SlopeCalculating Slope
• Slope = m = Slope = m = yy11 – y – y22 = = yy22 – y – y11
xx11 – x – x22 x x22 – x – x11
(5, 4) (2, 1)(5, 4) (2, 1)
• 4 - 14 - 1
• 5 - 2 5 - 2
• 33
• 33
• m = 1m = 1
(x1,y1) (x2,y2)
Ex # 2Ex # 2 A = (-4, 2) B=(2, -4)
(x1,y1) (x2,y2) -4 – 2
2 - - 4
- 6
6
m = -1
1 2 3 4 5-1-2-3-4-5
1
2
3
4
5
-1
-2
-3
-4
-5
x
y (4, 5)
(1, 1)
Ex #3Ex #3
(x2,y2)
(x1,y1)21
21
xx
yyslope
1 – 5
1 – 4
m = -4
-3
Two Decimal Place PracticeTwo Decimal Place Practice
• Find the slope of (4,217.08) Find the slope of (4,217.08) (11,575.41)(11,575.41)
• m = m = 217.08 – 575.41217.08 – 575.41
4-114-11
• 51.1951.19
Word ProblemsWord Problems
BASICSBASICS
• Set your x and ySet your x and y
• Find the linear function meaning find Find the linear function meaning find slope – m and y int or bslope – m and y int or b
• Solve questionSolve question
• x is always timex is always time
• y is always moneyy is always money
Word Problems Con’tWord Problems Con’t
KEYSKEYS
• PerPer
• RateRatem
•Fixed
•Initial b
Point – two values
point
Point – Point Point – Point
• Little Billy rents a car for 5 days and Little Billy rents a car for 5 days and pays $463.65. Little Sally rents a car pays $463.65. Little Sally rents a car for 14 days and pays $1208.22for 14 days and pays $1208.22
a)a) How much would it cost to rent a car How much would it cost to rent a car for 27 days?for 27 days?
b)b) How many days can you get for How many days can you get for $2531.90$2531.90
Step 1: Set Your x and y… what are Step 1: Set Your x and y… what are they?they?
SolutionSolution
• x is # of daysx is # of days
• y is $ (money)y is $ (money)
• (5, 463.65) (14, 1208.22)(5, 463.65) (14, 1208.22)
• m = m = 463.65 – 1208.22463.65 – 1208.22
5 – 145 – 14
• m = 82.73m = 82.73
• y = 82.73x + by = 82.73x + b
Solution Con’tSolution Con’t
• (5, 463.65) (5, 463.65) 463.65 = 82.73(5) + b 463.65 = 82.73(5) + b
• 463.65 = 413.65 + b463.65 = 413.65 + b
• 50 = b50 = b
• b = 50b = 50
• y = 82.73x + 50y = 82.73x + 50
• A) x = 27 A) x = 27 y = 82.73 (27) + 50 y = 82.73 (27) + 50
• y = $2283.71y = $2283.71
Solution Con’tSolution Con’t
• b) y = 2531.90 b) y = 2531.90 2531.90 = 82.73x 2531.90 = 82.73x + 50+ 50
• 2481.90 = 82.73x2481.90 = 82.73x
• 30 = x 30 = x
• x = 30x = 30
Slope Point Slope Point
• Little Billy rents a car and pays a rate Little Billy rents a car and pays a rate of $43.91 per day. Little Sally rents of $43.91 per day. Little Sally rents for 42 days and pays $1900.45for 42 days and pays $1900.45
• a) How much would it cost to rent a a) How much would it cost to rent a car for 27 days?car for 27 days?
• b) How many days can you get for b) How many days can you get for $2471.28?$2471.28?
• Step 1: Set your x and yStep 1: Set your x and y
SolutionSolution
• x is # of daysx 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) (42,1900.45) 1900.45 = 43.91 (42) + b 1900.45 = 43.91 (42) + b
• 1900.45 = 1844.22 + b1900.45 = 1844.22 + b
• 56.23 = b56.23 = b
• b = 56.23b = 56.23
Solution Con’tSolution Con’t
• y = 43.91x + 56.23y = 43.91x + 56.23
• A) x = 27 A) x = 27 y = 43.91 (27) + 56.23 y = 43.91 (27) + 56.23
• y = $1241.80y = $1241.80
• B) y = 2471.28 B) y = 2471.28 2471.28=4391 2471.28=4391 x+56.23x+56.23
• 2415.05 = 43.91x2415.05 = 43.91x
• 55 = x 55 = x
• x = 55x = 55
Y Int - PointY Int - Point
• Little Billy rents a car and pays a fixed cost Little Billy rents a car and pays a fixed cost of $132.11. Little Sally rents for 12 days of $132.11. Little Sally rents for 12 days and pays $1331.99and 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 daysx is # of days• y is $ moneyy is $ money• b = 132.11 (12, 1331.99)b = 132.11 (12, 1331.99)• y = mx + 132.11y = mx + 132.11
Solution Solution
• (12, 1331.99) (12, 1331.99) 1331.99 =12m+132.11 1331.99 =12m+132.11
• 1199.88 = 12m1199.88 = 12m
• 99.99 = m 99.99 = m
• m = 99.99m = 99.99
• y = 99.99x + 132.11y = 99.99x + 132.11
• a) x = 27 a) x = 27 y = 99.99 (27) + 132.11 y = 99.99 (27) + 132.11
• y = $2831.84y = $2831.84
Solution (Con’t) Solution (Con’t)
• B) y = 1232 B) y = 1232 1232 = 99.99x + 1232 = 99.99x + 132.11132.11
• 1099.89 = 99.99x1099.89 = 99.99x
• 11 = x 11 = x
• x = 11x = 11
Slope – Y intSlope – Y int
• Little Billy rents a car and pays a rate of Little Billy rents a car and pays a rate of $73.99 per day. Little Sally pays a fixed cost $73.99 per day. Little Sally pays a fixed cost of $42.19of $42.19
• a) How much for 27 days?a) How much for 27 days?
• b) How many days for $412.14b) How many days for $412.14
• x is # of daysx is # of days
• y is $ (money)y is $ (money)
• m = 73.99 b = 42.99 m = 73.99 b = 42.99
• y = 73.99x + 42.19y = 73.99x + 42.19
Solutions Solutions
• x= 27 x= 27 y = 73.99 (27) + 42.19 y = 73.99 (27) + 42.19
• y = 2039.92y = 2039.92
• b) y = 412.14 b) y = 412.14 412.14 = 412.14 = 73.99x+42.1973.99x+42.19
• 369.95 = 73.99x369.95 = 73.99x
• 5 = x5 = x
• x = 5x = 5
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