...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