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