REVIEW: SLOPE OF A LINE What is slope of a line? --how to move from one point on the line to another...

REVIEW: SLOPE OF A LINEWhat is slope of a

line?--how to move from one point on the line to

another point on the line

AB

CD

--these moves are classified as rise (up-down)

move over run(right-left)

move

run3

rise 2run3 rise 2

run3 rise

2

The slope between any two points on the same line is always the same.

REVIEW: SLOPE OF A LINEWhat is slope of a

line?--The coordinates of two points on

a line can be used to find the slope of the line by the formula:

AB

CD

(- 2, 0)

(1, 2)(4, 4)

(7, 6)

12

12

xxyy

m

EXAMPLE: (using points A and B)

32

212

)2(102

m

rise

run

5, 7, 9, 11, 13, 15, 17

+2 +2 +2 +2 +2 +2

USING SLOPE:NEXT/NOW STATEMENTS

Consider the following number sequence:

Value (y) change

Position (x):

0 1 2 3 4 5 6

Starting Position

212)( valuestwoanybetweenslope

+1 +1 +1 +1 +1 +1

Starting Value


1 2 3 4 5 6

5

7

9

11

13

15

17

0

SLOPE-INTERCEPTEQUATION OF A LINE:

y = m x + b

slope

In this problem:y = 2 x + 5

y-Intercept

slope y-Intercept

(starting value)


1 2 3 4 5 6

5

7

9

11

13

15

17

0

SLOPE-INTERCEPTEQUATION OF A LINE:

y = slope (x) + y- intercept

Another way to compute the value (y):

NEXT (y value) = NOW (present y value) +

SLOPE; STARTING AT Y-INTERCEPT


1 2 3 4 5 6

5

7

9

11

13

15

17

0

NEXT (y value) = NOW (present y value) +

SLOPE; STARTING AT Y-INTERCEPT

EXAMPLE:Starting at a y-value of 9 (NOW)Add the slope (2)9 + 2 = 11 which is the NEXT y-value

Initial Value 1st 2nd 3rd 4th

There are two inches on the ground at midnight. Each hourthe snowfall increase by ½ inch.

SITUATION 1 Snowfall

X y

TABLE

GRAPH

SEQUENCE

SLOPE (rate of change)

Y-intercept (initial value)

NEXT-NOW STATEMENT Practical Domain

Practical Range

EQUATION

21/2

Y = 1/2x + 2 Hour after midnight

Snowfall (inches)

012

3

2.52

33.5

+1/2

+1/2

2+1/2 +1/2 +1/2 +1/2

2.5 3 3.5 4

NEXT = NOW + ½; starting at 2All non-negative numbers

All numbers at least two

12 1

2

+1/2

There are 6 inches of rain on the ground at midnight. It keeps falling at ¼ inch per hour.

SITUATION 2 Rainfall

X yTABLE

GRAPH

SEQUENCEInitial Value

1st 2nd 3rd 4th



NEXT-NOW STATEMENT Practical DomainPractical Range

EQUATION

y = ¼ x + 6

¼ 6 Hours aftermidnight

rainfall(inches)

NEXT = NOW + ¼; STARTING AT 6

0

6

1

6.25

2

6.50

3

6.75

4

7.00

6

6.25

6.50

6.75

7.00

x ≥ 0

y ≥ 6

SITUATION 3 (Cab fare)

XMiles

YCab fare

0 31 5.502 83 10.5

TABLE

GRAPH


1st 2nd 3rd 4th




Practical Range

EQUATION

A cab company charges a $ 3 flat rate plus $ 2.50 per mile.

2.5 3

y = 2.5 x + 3

3

5.50

8

10.50

13

NEXT = NOW + 2.5; STARTING AT 3

x ≥ 0

y ≥ 3

SITUATION 4 (money saved in bank)

X yTABLE

GRAPH


1st 2nd 3rd 4th

50 70 90 110




EQUATION

You have a beginning balance of $ 30. You add $ 20 every week.

Weeks of

saving

Amount

20 30

y = 20 x + 30 0

30

1

50

2

70

3

90

4

110

30

NEXT = NOW + 20; STARTING AT 30

x ≥ 0

y ≥ 30

SITUATION 5 (money in wallet paying for sodas)

X yTABLE

GRAPH


1st 2nd 3rd 4th

8 6 4 2




Practical Range

EQUATION

John has $ 10 in his wallet. Each soda he buys costs $ 2.00.

- 2 10

10

Numberof

sodasbought

Amountleft

0

10

1

8

2

6

3

4

4

2

y = - 2 x + 10

0

1

2

3

4

5

6

7

8

9

10

11

1

2

3

4

5

6

7

8

10

11

9

NEXT = NOW – 2 ; STARTING AT 10

0 ≤ x ≤ 5

0 ≤ y ≤ 10

SITUATION 6 (Water in bathtub)

X yTABLE

GRAPH

SEQUENCE



EQUATION 4

- 2

Slope = - 2/4 = - ½

A bathtub has 2 gallons of water until the plug is pulled. It then loses ½ gallon per minute. - ½

Initial Value

1st 2nd 3rd 4th

Y-intercept (initial value) 2

Numberof

hours after

Amountleft

0

2

1

1.5

2

1.0

3

0.5

4

0

y = - ½ x + 2

2 1.5 1.0 0.5 0

NEXT = NOW – ½ ; STARTING AT 2

0 ≤ x ≤ 4

0 ≤ y ≤ 2

SITUATION 7 (Filling a bathtub)

X yTABLE

GRAPH


1st 2nd 3rd 4th




EQUATION

Next = Now + ½; starting at 3

A bathtub has 3 gallons of water. The water is turned on and it fills at a rate of ½ gallon per minute. ½ 3

Numberof

hours after

Amount

0

3

1

3.5

2 4.0

3 4.5

4 5.0

y = ½ x + 3

3 3.5 4.0 4.5 5.0 x ≥ 0

y ≥ 3

SITUATION 8

X Time (min)

Y Words typed

2 48

4 96

TABLE

GRAPH


1st 2nd 3rd 4th




EQUATION

Slope = (96 – 48) / (4 – 2) = 48/2 = 24

Sue can type 24 words per minute. 24 0

0

0

1

24

3

72

0 24 48 72 96

y = 24 x

NEXT = NOW + 24 ; STARTING AT 0

x ≥ 0

y ≥ 0

SITUATION 9 (Lemonade Stand)

X yTABLE

GRAPH


1st 2nd 3rd 4th

SLOPE 2(rate of change)

Y-intercept -10 (initial value)


Practical Range

EQUATION 0

1

2

3

4

5

6

7

8

9

10

11

-1

-3

-5

-7

-9

-11

-2

-4

-6

-8

-10

1

2

3

4

5

6

7

Sally spends $ 10 for lemonade supplies. She charges $ 2 per cup.

Number

of cups

Profit

0

- 10

1

- 8

2

- 6

3

- 4

4

- 2

- 10 - 8 - 6 - 4 - 2

y = 2 x – 10

NEXT = NOW + 2 ; STARTING AT - 10

x ≥ 0

y ≥ -10

SITUATION 10 (Waiting in Line)

X yTABLE

GRAPH


1st 2nd 3rd 4th




Practical Range

EQUATION

5

10

15

20

25

30

35

40

45

50

There are 10 people waiting in line before a store opens. Five people enter the line every two minutes. 5/2 10

Time(min)

people

0

10

1

12.5

2

15

3

17.5

4

20

10 12.5 15 17.5 20

y = 2 .5x + 10

NEXT = NOW + 2 ; STARTING AT 10

x ≥ 0

y ≥ 10

SITUATION 11 (Temperature (Co) over time)

X yTABLE

GRAPH


1st 2nd 3rd 4th

7 3 -1




Practical Range

EQUATION

1

5

7

8

10

0

1

-1

2

2

-2

3

3

-3

4

4

-4

5

-5

6

6

-6

7

-7

8

9

9

10

11

11

12

The temperature is 11 ° initially and drops 4 ° every hour. - 4 11

Time(hr)

degrees

0

11

1

7

2

3

3

- 1

4

- 5

10 - 5

y = - 4 x + 11

NEXT = NOW – 4 ; STARTING AT 11

x ≥ 0

y ≤ 11

REVIEW: SLOPE OF A LINE What is slope of a line? --how to move from one point on the line to another...

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Transcript of REVIEW: SLOPE OF A LINE What is slope of a line? --how to move from one point on the line to another...