AA Section 3-5
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Transcript of AA Section 3-5
...or the Point-Slope Theorem section
Section 3-5Finding the Equation of a Line
Warm-upWrite an equation for the line through the pair of points
a. (5, 9), (5, -2) b. (9, 1), (6, 4)
Warm-upWrite an equation for the line through the pair of points
a. (5, 9), (5, -2) b. (9, 1), (6, 4)
x = 5
Warm-upWrite an equation for the line through the pair of points
a. (5, 9), (5, -2) b. (9, 1), (6, 4)
x = 5 y = -x + 10
Warm-upWrite an equation for the line through the pair of points
a. (5, 9), (5, -2) b. (9, 1), (6, 4)
x = 5 y = -x + 10
Not quite sure how this is done? We’ll see two ways today
Question
What determines a line?
Question
What determines a line?
Two points
Question
What determines a line?
Two points
...and once we have two points, we can find an equation
Example 1The formula relating blood pressure and age is linear.
Normal systolic blood pressures are 110 for a 20 year old and 130 for a 60 year old. Graph the line and find an
equation where blood pressure B is a function of age A.
Example 1The formula relating blood pressure and age is linear.
Normal systolic blood pressures are 110 for a 20 year old and 130 for a 60 year old. Graph the line and find an
equation where blood pressure B is a function of age A.
0
32.5
65.0
97.5
130.0
0 15 30 45 60
Blo
od P
ress
ure
Age
Example 1 (con’t)(20, 110), (60, 130)
Example 1 (con’t)(20, 110), (60, 130)
m =
130− 11060−20
Example 1 (con’t)(20, 110), (60, 130)
m =
130− 11060−20
=2040
Example 1 (con’t)(20, 110), (60, 130)
m =
130− 11060−20
=2040
=12
Example 1 (con’t)(20, 110), (60, 130)
m =
130− 11060−20
=2040
=12
B = 12
A+b
Example 1 (con’t)(20, 110), (60, 130)
m =
130− 11060−20
=2040
=12
110 = 12(20)+b
B = 12
A+b
Example 1 (con’t)(20, 110), (60, 130)
m =
130− 11060−20
=2040
=12
110 = 12(20)+b
110 = 10+b
B = 12
A+b
Example 1 (con’t)(20, 110), (60, 130)
m =
130− 11060−20
=2040
=12
110 = 12(20)+b
110 = 10+b b = 100
B = 12
A+b
Example 1 (con’t)(20, 110), (60, 130)
m =
130− 11060−20
=2040
=12
110 = 12(20)+b
110 = 10+b b = 100
B = 12
A+ 100
B = 12
A+b
Example 1 (con’t)(20, 110), (60, 130)
m =
130− 11060−20
=2040
=12
110 = 12(20)+b
110 = 10+b b = 100
B = 12
A+ 100
B = 12
A+b
There has to be a better way!
Point-Slope Theorem
Point-Slope Theorem
If a line contains (x1, y1) and has slope m, then it has the equation y - y1 = m(x - x1)
Point-Slope Theorem
If a line contains (x1, y1) and has slope m, then it has the equation y - y1 = m(x - x1)
(In other words, you need a point and the slope)
Example 2Find an equation for the line through (-3, 6) and (5, 0)
using the point-slope theorem.
Example 2Find an equation for the line through (-3, 6) and (5, 0)
using the point-slope theorem.
m =
6−0−3− 5
Example 2Find an equation for the line through (-3, 6) and (5, 0)
using the point-slope theorem.
m =
6−0−3− 5
=6−8
Example 2Find an equation for the line through (-3, 6) and (5, 0)
using the point-slope theorem.
m =
6−0−3− 5
=6−8
= −34
Example 2Find an equation for the line through (-3, 6) and (5, 0)
using the point-slope theorem.
m =
6−0−3− 5
=6−8
= −34
y − y1 = m(x − x1 )
Example 2Find an equation for the line through (-3, 6) and (5, 0)
using the point-slope theorem.
m =
6−0−3− 5
=6−8
= −34
y − y1 = m(x − x1 )
y −0 = − 3
4(x − 5)
Example 2Find an equation for the line through (-3, 6) and (5, 0)
using the point-slope theorem.
m =
6−0−3− 5
=6−8
= −34
y − y1 = m(x − x1 )
y −0 = − 3
4(x − 5)
y = − 3
4x + 15
4
When dealing with real world situations, deal with the problem as we always have: find the equation first, then
answer the question.
Example 3The lightest recommended weight for a Martian with height 4’10” is 109 lbs. This weight increases 2 lbs/in to a height of 5’1” and then goes up 3 lbs/in to a height
of 6’ which is tall for a Martian.a. Graph the situation
Example 3The lightest recommended weight for a Martian with height 4’10” is 109 lbs. This weight increases 2 lbs/in to a height of 5’1” and then goes up 3 lbs/in to a height
of 6’ which is tall for a Martian.a. Graph the situation
First, we need to convert all heights to inches
Example 3The lightest recommended weight for a Martian with height 4’10” is 109 lbs. This weight increases 2 lbs/in to a height of 5’1” and then goes up 3 lbs/in to a height
of 6’ which is tall for a Martian.a. Graph the situation
First, we need to convert all heights to inches4’10” =
Example 3The lightest recommended weight for a Martian with height 4’10” is 109 lbs. This weight increases 2 lbs/in to a height of 5’1” and then goes up 3 lbs/in to a height
of 6’ which is tall for a Martian.a. Graph the situation
First, we need to convert all heights to inches4’10” = 4(12) +10 =
Example 3The lightest recommended weight for a Martian with height 4’10” is 109 lbs. This weight increases 2 lbs/in to a height of 5’1” and then goes up 3 lbs/in to a height
of 6’ which is tall for a Martian.a. Graph the situation
First, we need to convert all heights to inches4’10” = 4(12) +10 = 58”
Example 3The lightest recommended weight for a Martian with height 4’10” is 109 lbs. This weight increases 2 lbs/in to a height of 5’1” and then goes up 3 lbs/in to a height
of 6’ which is tall for a Martian.a. Graph the situation
First, we need to convert all heights to inches4’10” = 4(12) +10 = 58”5’1” =
Example 3The lightest recommended weight for a Martian with height 4’10” is 109 lbs. This weight increases 2 lbs/in to a height of 5’1” and then goes up 3 lbs/in to a height
of 6’ which is tall for a Martian.a. Graph the situation
First, we need to convert all heights to inches4’10” = 4(12) +10 = 58”5’1” = 5(12) + 1 =
Example 3The lightest recommended weight for a Martian with height 4’10” is 109 lbs. This weight increases 2 lbs/in to a height of 5’1” and then goes up 3 lbs/in to a height
of 6’ which is tall for a Martian.a. Graph the situation
First, we need to convert all heights to inches4’10” = 4(12) +10 = 58”5’1” = 5(12) + 1 = 61”
Example 3The lightest recommended weight for a Martian with height 4’10” is 109 lbs. This weight increases 2 lbs/in to a height of 5’1” and then goes up 3 lbs/in to a height
of 6’ which is tall for a Martian.a. Graph the situation
First, we need to convert all heights to inches4’10” = 4(12) +10 = 58”5’1” = 5(12) + 1 = 61”
6’ = 6(12) = 72”
Example 3The lightest recommended weight for a Martian with height 4’10” is 109 lbs. This weight increases 2 lbs/in to a height of 5’1” and then goes up 3 lbs/in to a height
of 6’ which is tall for a Martian.a. Graph the situation
First, we need to convert all heights to inches4’10” = 4(12) +10 = 58”5’1” = 5(12) + 1 = 61”
6’ = 6(12) = 72”
(58”, 109 lbs)
Example 3The lightest recommended weight for a Martian with height 4’10” is 109 lbs. This weight increases 2 lbs/in to a height of 5’1” and then goes up 3 lbs/in to a height
of 6’ which is tall for a Martian.a. Graph the situation
First, we need to convert all heights to inches4’10” = 4(12) +10 = 58”5’1” = 5(12) + 1 = 61”
6’ = 6(12) = 72”
(58”, 109 lbs) (61”, 115 lbs)
Example 3The lightest recommended weight for a Martian with height 4’10” is 109 lbs. This weight increases 2 lbs/in to a height of 5’1” and then goes up 3 lbs/in to a height
of 6’ which is tall for a Martian.a. Graph the situation
First, we need to convert all heights to inches4’10” = 4(12) +10 = 58”5’1” = 5(12) + 1 = 61”
6’ = 6(12) = 72”
(58”, 109 lbs) (61”, 115 lbs) (72”, 148 lbs)
Example 3 (con’t)(58”, 109 lbs) (61”, 115 lbs) (72”, 148 lbs)b. Find two equations that describe these situations
Example 3 (con’t)(58”, 109 lbs) (61”, 115 lbs) (72”, 148 lbs)b. Find two equations that describe these situations
y - y1 = m(x - x1)
Example 3 (con’t)(58”, 109 lbs) (61”, 115 lbs) (72”, 148 lbs)b. Find two equations that describe these situations
y - y1 = m(x - x1)w - w1 = m(h - h1)
Example 3 (con’t)(58”, 109 lbs) (61”, 115 lbs) (72”, 148 lbs)b. Find two equations that describe these situations
y - y1 = m(x - x1)
w - 109 = 2(h - 58)w - w1 = m(h - h1)
Example 3 (con’t)(58”, 109 lbs) (61”, 115 lbs) (72”, 148 lbs)b. Find two equations that describe these situations
y - y1 = m(x - x1)
w - 109 = 2(h - 58)w - w1 = m(h - h1)
w = 2h - 7
Example 3 (con’t)(58”, 109 lbs) (61”, 115 lbs) (72”, 148 lbs)b. Find two equations that describe these situations
y - y1 = m(x - x1)
w - 109 = 2(h - 58)w - w1 = m(h - h1)
w = 2h - 7for 58 ≤ h ≤ 61
Example 3 (con’t)(58”, 109 lbs) (61”, 115 lbs) (72”, 148 lbs)b. Find two equations that describe these situations
y - y1 = m(x - x1)
w - 109 = 2(h - 58)w - w1 = m(h - h1)
w = 2h - 7for 58 ≤ h ≤ 61
w - w1 = m(h - h1)
Example 3 (con’t)(58”, 109 lbs) (61”, 115 lbs) (72”, 148 lbs)b. Find two equations that describe these situations
y - y1 = m(x - x1)
w - 109 = 2(h - 58)w - w1 = m(h - h1)
w = 2h - 7for 58 ≤ h ≤ 61
w - w1 = m(h - h1)w - 115 = 3(h - 61)
Example 3 (con’t)(58”, 109 lbs) (61”, 115 lbs) (72”, 148 lbs)b. Find two equations that describe these situations
y - y1 = m(x - x1)
w - 109 = 2(h - 58)w - w1 = m(h - h1)
w = 2h - 7for 58 ≤ h ≤ 61
w - w1 = m(h - h1)w - 115 = 3(h - 61)
w = 3h - 68
Example 3 (con’t)(58”, 109 lbs) (61”, 115 lbs) (72”, 148 lbs)b. Find two equations that describe these situations
y - y1 = m(x - x1)
w - 109 = 2(h - 58)w - w1 = m(h - h1)
w = 2h - 7for 58 ≤ h ≤ 61
w - w1 = m(h - h1)w - 115 = 3(h - 61)
w = 3h - 68for 61 < h ≤ 72
Homework
Homework
p. 165 #1 - 21
“ I’m not sure I want popular opinion on my side -- I’ve noticed those with the most opinions often have the
fewest facts.” - Bethania McKenstry