Download initial value

REVIEW: SLOPE OF A LINE
What is slope of a line?
--how to move from
D
one point on the line
2
C 3 rise
to another point on
run
the line
B run rise 2
3
--these moves are
rise 2
A
classified as rise
(up-down) move
run
over run(right-left)
3
move
The slope between any two points on the same
line is always the same.
REVIEW: SLOPE OF A LINE
What 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:
y2  y1
m
x2  x1
EXAMPLE: (using
points A and B)
20
2
2
m


1  ( 2) 1  2 3
D
C
B
(1, 2)
A
(- 2, 0)
rise
run
(7, 6)
(4, 4)
USING SLOPE:
NEXT/NOW STATEMENTS
Consider the following number sequence:
Starting Value
+2
+2
+2
+2
+2
+2
Value (y) change
Position (x):
Starting Position
5, 7, 9, 11, 13, 15, 17
0
+1
1
2
+1
4
3
+1
+1
6
5
+1
+1
2
slope (between any two values)   2
1
USING SLOPE:
NEXT/NOW STATEMENTS
SLOPE-INTERCEPT
EQUATION OF A LINE:
y=mx+b
slope
yIntercept
In this problem:
y=2x+5
slope
yIntercept
(starting value)
17
15
13
11
9
7
5
0
1 2 3 4 5 6
USING SLOPE:
NEXT/NOW STATEMENTS
SLOPE-INTERCEPT
EQUATION 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
17
15
13
11
9
7
5
0
1 2 3 4 5 6
USING SLOPE:
NEXT/NOW STATEMENTS
NEXT (y value) = NOW
(present y value) + SLOPE;
STARTING AT Y-INTERCEPT
17
15
13
11
EXAMPLE:
9
Starting at a yvalue of 9 (NOW)
7
Add the slope (2)
5
9 + 2 = 11 which is
the NEXT y-value
0
1 2 3 4 5 6
SITUATION 1
Snowfall
There are two inches on the ground at midnight. Each hour
the snowfall increase by ½ inch.
SLOPE
Y-intercept
1/
2
(rate of change)
2 +1/
2
2.5 +1/
2
3 +1/
2
3.5
3
2 1
EQUATION
X Hour
y Snowfall
after midnight (inches)
0
1
2
2
2
(initial value)
TABLE
GRAPH
1
Y = 1 /2 x + 2
SEQUENCE
Initial Value
2
1st
+1/2
2.5
2nd
+1/2
3
3rd
+1/2
4th
3.5
+1/2
4
NEXT-NOW STATEMENT
Practical Domain
NEXT = NOW + ½; starting at 2
Practical Range
All non-negative numbers
All numbers at least two
SITUATION 2 Rainfall
There are 6 inches of rain on the ground at midnight.
It keeps falling at ¼ inch per hour.
SLOPE
(rate of change)
¼
Y-intercept
(initial value)
TABLE
Hours
GRAPH
6
EQUATION
X after
y rainfall
midnight (inches)
0
1
2
3
4
6
6.25
6.50
6.75
7.00
y= ¼x+6
SEQUENCE
Initial Value
6
1st
6.25
2nd
6.50
NEXT-NOW STATEMENT
3rd
6.75
4th
7.00
x≥0
Practical
Domain
NEXT = NOW + ¼; STARTING AT 6
Practical Range y ≥ 6
SITUATION 3
(Cab fare)
A cab company charges a $ 3 flat rate plus $ 2.50 per
mile.
SLOPE
2.5
(rate of change)
Y-intercept
(initial value)
TABLE
GRAPH
3
EQUATION
X
Miles
Y
Cab fare
0
3
1
5.50
2
8
3
10.5
y = 2.5 x + 3
SEQUENCE
Initial Value
3
1st
5.50
2nd
8
3rd
4th
10.50
13
NEXT-NOW STATEMENT
NEXT = NOW + 2.5;
STARTING AT 3
Practical Domain x ≥ 0
Practical Range
y≥3
SITUATION 4
(money saved in bank)
You have a beginning balance of $ 30. You add $ 20
every week.
SLOPE
(rate of change)
TABLE
Weeks
20
Y-intercept
(initial value)
GRAPH
30
EQUATION
X of
saving
y
Amount
0
1
2
3
4
30
50
70
90
110
y = 20 x + 30
SEQUENCE
Initial Value
30
1st
2nd
3rd
4th
50
70
90
110
NEXT-NOW STATEMENT
NEXT = NOW + 20;
STARTING AT 30
Practical Domain x ≥ 0
Practical Range
y ≥ 30
SITUATION 5 (money in wallet paying for sodas)
John has $ 10 in his wallet. Each soda he buys costs
$ 2.00.
SLOPE
(rate of change)
Y-intercept
-2
(initial value)
TABLE
Number
EQUATION
X of sodas yAmount
bought
left
0
1
2
3
4
10
10
8
6
4
2
y = - 2 x + 10
SEQUENCE
Initial Value
10
1st
2nd
3rd
4th
8
6
4
2
NEXT-NOW STATEMENT
NEXT = NOW – 2 ;
STARTING AT 10
GRAPH
11
10
9
8
7
6
5
4
3
2
1
0 1 2 3 4 5 6 7 8 9 10 11
Practical Domain 0 ≤ x ≤ 5
Practical Range 0 ≤ y ≤ 10
SITUATION 6 (Water in bathtub)
A bathtub has 2 gallons of water until the plug is
pulled. It then loses ½ gallon per minute.
SLOPE
(rate of change)
-½
Y-intercept
(initial value)
TABLE
Number
X of hours y Amount
left
after
0
1
2
3
4
2
1.5
1.0
0.5
0
2
EQUATION
4
y=-½ x+2
-2
Slope =
- 2 /4 = - ½
SEQUENCE
Initial Value
2
1st
1.5
GRAPH
2nd
1.0
NEXT-NOW STATEMENT
NEXT = NOW – ½ ;
STARTING AT 2
3rd
0.5
4th
0
Practical Domain 0 ≤ x ≤ 4
Practical Range 0 ≤ y ≤ 2
SITUATION 7 (Filling a bathtub)
A bathtub has 3 gallons of water. The water is turned
on and it fills at a rate of ½ gallon per minute.
SLOPE
(rate of change)
½
TABLE
Number
X of hours y
Amount
after
0
1
2
3
4
3
3.5
4.0
4.5
5.0
Y-intercept
(initial value)
GRAPH
3
EQUATION
y=½ x+3
SEQUENCE
Initial Value
3
1st
2nd
3.5
4.0
3rd
4.5
NEXT-NOW STATEMENT
Next = Now + ½; starting at
3
4th
5.0
Practical Domain
x≥0
Practical Range y ≥ 3
SITUATION 8
Sue can type 24 words per minute.
SLOPE
24
(rate of change)
TABLE
X Time
(min)
Y-intercept
GRAPH
0
(initial value)
EQUATION
Y
Words
typed
0
1
0
24
2
48
3
72
4
96
y = 24 x
Slope = (96 – 48) / (4 – 2) =
48/ = 24
2
SEQUENCE
Initial Value
0
1st
24
2nd
48
NEXT-NOW STATEMENT
NEXT = NOW + 24 ;
STARTING AT 0
3rd
72
4th
96
Practical Domain
Practical Range
x≥0
y≥0
SITUATION 9 (Lemonade Stand)
Sally spends $ 10 for lemonade
supplies. She charges $ 2 per cup.
SLOPE
2
(rate of change)
Y-intercept -10
(initial value)
TABLE
EQUATION
X Number y
Profit
of cups
0
1
2
3
4
- 10
-8
-6
-4
-2
y = 2 x – 10
SEQUENCE
Initial
Value
- 10
1st
-8
2nd
-6
3rd
-4
NEXT-NOW STATEMENT
NEXT = NOW + 2 ;
STARTING AT - 10
4th
-2
GRAPH
7
6
5
4
3
2
1
0
-1 1 2 3 4 5 6 7 8 9 1011
-2
-3
-4
-5
-6
-7
-8
-9
-10
-11
Practical Domain x ≥ 0
Practical Range y ≥ -10
SITUATION 10 (Waiting in Line)
There are 10 people waiting in line before a store opens.
Five people enter the line every two minutes.
SLOPE
(rate of change)
Y-intercept
5/2 (initial value)
TABLE
X
Time
(min)
0
1
2
3
4
50
45
10
40
35
EQUATION
y
people
10
12.5
15
17.5
20
GRAPH
30
y = 2 .5x + 10
25
20
15
10
5
SEQUENCE
Initial Value
10
1st
12.5
2nd
15
3rd
NEXT-NOW STATEMENT
NEXT = NOW + 2 ;
STARTING AT 10
17.5
4th
20
Practical Domain x ≥ 0
Practical Range y ≥ 10
SITUATION 11 (Temperature (Co) over time)
The temperature is 11 ° initially and
drops 4 ° every hour.
SLOPE
(rate of change)
-4
Y-intercept
(initial value)
TABLE
X
Time
(hr)
0
1
2
3
4
11
EQUATION
y
degrees
11
7
3
-1
-5
y = - 4 x + 11
SEQUENCE
Initial
Value
10
1st
2nd
3rd
7
3
-1
NEXT-NOW STATEMENT
NEXT = NOW – 4 ;
STARTING AT 11
4th
-5
11
10
9
8
7
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
GRAPH
1 2 3 4 5 6 7 8 9 101112
Practical Domain x ≥ 0
Practical Range y ≤ 11