Transcript Chapter Two 2.5
§ 2.5
The Point-Slope Form of the Equation of a Line
Point-Slope Form
144
Point-Slope Form of the Equation of a Line
The point-slope equation of a nonvertical line with slope
m
1 1
y
y
1
m
x
x
1 Blitzer,
Intermediate Algebra
, 5e – Slide #2 Section 2.5
Point-Slope Form
145 EXAMPLE Write the point-slope form and then the slope-intercept form of the equation of the line with slope -3 that passes through the point (2,-4).
SOLUTION
y
y
y
1
y
4
m
3 3
x x x
x
1 2 2 Substitute the given values Simplify This is the equation of the line in
point-slope form
.
y
4
y
3
x
3
x
6 2 Distribute Subtract 4 from both sides This is the equation of the line in
slope-intercept form
.
Blitzer,
Intermediate Algebra
, 5e – Slide #3 Section 2.5
Point-Slope Form
145 EXAMPLE Write the point-slope form and then the slope-intercept form of the equation of the line that passes through the points (2,-4) and (-3,6).
SOLUTION First I must find the slope of the line. That is done as follows:
m
6 3 2 10 5 2 Blitzer,
Intermediate Algebra
, 5e – Slide #4 Section 2.5
Point-Slope Form
145 CONTINUED Now I can find the two forms of the equation of the line. In find the point-slope form of the line, I can use either point provided. I’ll use (2,-4).
y
y y
y
1 4
m
2 2
x x x
x
1 2 2 Substitute the given values Simplify This is the equation of the line in
point-slope form
.
y
4
y
2 2
x x
4 Distribute Subtract 4 from both sides This is the equation of the line in
slope-intercept form
.
Blitzer,
Intermediate Algebra
, 5e – Slide #5 Section 2.5
Equations of Lines
146
Equations of Lines
Standard Form Slope-Intercept Form Horizontal Line Vertical Line Point-slope Form
Ax
+
By
=
C y
=
mx
+
b y
=
b y
y
1
x
=
a m
x
x
1 Blitzer,
Intermediate Algebra
, 5e – Slide #6 Section 2.5
Deciding which form to use:
146
y = mx + b
Begin with the slope-intercept form if you know: The slope of the line and the y intercept or Two points on the line, one of which is the y –intercept
y – y
1
= m(x – x
1
)
Begin with the point-slope form if you know: The slope of the line and a point on the line other than the y intercept or Two points on the line, neither of which is the y-intercept Blitzer,
Intermediate Algebra
, 5e – Slide #7 Section 2.5
Modeling Life Expectancy
147 EXAMPLE 3 Go over example 3 See Figures 2.35(a) and 2.35(b) Use the data points to write the slope-intercept form of the equation of this line. Use the linear function to predict the life expectancy of an American man born in 2020.
Find the slope of 0.215. The slope indicates that for each subsequent birth year, a man’s life expectancy is increasing by 0.215 years.
Use the point-slope form to write the equation (model).
y – 70.0 = 0.215(x – 20) Change to slope-intercept form y = 0.215x + 65.7
Life expectancy of men born in 2020 f(60) = 0.215(60) + 65.7 = 78.6 years Blitzer,
Intermediate Algebra
, 5e – Slide #8 Section 2.5
Modeling Life Expectancy
148 Check Points Do Check Point 3 on page 148 See Figure 2.36
Use the data points to write the slope-intercept form of the equation of this line. (round to 2 decimal places). Use the linear function to predict the life expectancy of an American woman born in 2020.
Blitzer,
Intermediate Algebra
, 5e – Slide #9 Section 2.5
Modeling Life Expectancy
148 Check Points, continued Do Check Point 3 on page 148 See Figure 2.36
Use the data points to write the slope-intercept form of the equation of this line. (round to 2 decimal places). Use the linear function to predict the life expectancy of an American woman born in 2020.
Blitzer,
Intermediate Algebra
, 5e – Slide #10 Section 2.5
Modeling Public Tuition
Public Tuition: In 2005, the average cost of tuition and fees at public four-year colleges was $6130 , and in 2010 it was $7610 . Note that the known value for 2008 is $6530.
Solution: The line passes through (2005, 6.1) and (2010, 7.6). Find the slope.
m
change in y change in x 7610 6130 2010 2005 1480 5 296 Thus, the slope of the line is 296; tuition and fees on average increased by $296/yr.
Substitute 5 for 2005, 10 for 2010, and 8 for 2008.
m
change in y change in x 7610 6130 10 05 1480 5 296 Figure not in book
Modeling Public Tuition
Modeling public tuition: Write the slope-intercept form of the of the line shown in the graph. What is the y-intercept and does it have meaning in this situation.
y
y
y
1 6130
m
x
296
x
x
1 2005 1480 5 296 This is the equation of the line in
point-slope form
.
y
6130 296
x
593480
y
296
x
587350 This is the equation of the line in
slope-intercept form
.
Modeling public tuition: Substitute 5 for 2005, 10 for 2010, and 8 for 2008.
y y
y
1 6130
m
x
296
x
1
x
5 This is the equation of the line in
point-slope form
.
y
6130 296
x
1480
y
296
x
4650 This is the equation of the line in
slope-intercept form
.
Modeling Public Tuition
Using the slope-intercept form of the equation of the line shown in the graph. Use the equation to predict the average cost of tuition and fees at public four-year colleges in 2008.
y
296
x
587350
y
296 ( 2008 ) 587350 7018 Substitute 2008 or 8 for x and compute y.
y
296 ( 8 ) 4650 7018 The model predicts that the tuition in 2008 will be $7018 Use the equation to predict the average cost of tuition and fees at public four-year colleges in 2015.
Substitute 2015 or15 for x and compute y.
y
296 ( 2015 ) 587350
y
296 ( 15 ) 4650 9090 9090 The model predicts that the tuition in 2015 will be $9090.
Modeling the Graying of America
Write the slope-intercept form of the equation of the line shown in the graph. Use the equation to predict the median age of the U.S. population in 2020.
Solution: The line passes through (10, 30.0) and (30, 35.3). Find the slope.
m
change in y change in x 35 .
3 30 .
0 30 10 5 .
3 20 0 .
265 The slope indicates that each year the median age of the U.S. population is increasing by 0.265 year.
(30, 35.3) (10, 30.0)
Modeling the Graying of America
Write the slope-intercept form of the equation of the line shown in the graph. Use the equation to predict the median age of the U.S. population in 2020.
m
5 .
3 20 0 .
265 The slope indicates that each year the median age of the U.S. population is increasing by 0.265 year.
y y
y
1 30 .
0
m
x
0 .
265
x
1
x
10 This is the equation of the line in
point-slope form
.
y
30 .
0 0 .
265
x
2 .
65
y
0 .
265
x
27 .
35 This is the equation of the line in
slope-intercept form
.
A linear equation that models the median age of the U.S. population, y, x years after 1970.
(30, 35.3) (10, 30.0)
Modeling the Graying of America
Write the slope-intercept form of the equation of the line shown in the graph. Use the equation to predict the median age of the U.S. population in 2020.
m
5 .
3 20 0 .
265 The slope indicates that each year the median age of the U.S. population is increasing by 0.265 year.
y
0 .
265
x
27 .
35 A linear equation that models the median age of the U.S. population, y, x years after 1970.
Use the equation to predict the median age in 2020. Because 2020 is 50 years after 1970, substitute 50 for x and compute y.
y
0 .
265 ( 50 ) 27 .
35 40 .
6 The model predicts that the median age of the U.S. population in 2020 will be 40.6.
(30, 35.3) (10, 30.0)
Parallel and Perpendicular Lines
Slope and Parallel Lines
1) If two nonvertical lines are parallel, then they have the same slope.
2) If two
distinct
nonvertical lines have the same slope, then they are parallel.
3) Two
distinct
vertical lines, both with undefined slopes, are parallel.
y = 2x + 6 and y = 2x – 4 are parallel.
y = -4x +5 and y = -4x + 3 are parallel.
Blitzer,
Intermediate Algebra
, 5e – Slide #17 Section 2.5
Parallel and Perpendicular Lines
Slope and Perpendicular Lines
1) If two nonvertical lines are perpendicular, then the
product of their slopes
is -1.
2) If the product of the slopes of two lines is -1, then the lines are perpendicular.
3) A horizontal line having zero slope is perpendicular to a vertical line having undefined slope.
y = 2x + 6 and y = -(1/2)x – 4 are perpendicular.
y = -4x +5 and y = (1/4)x + 3 are perpendicular.
Blitzer,
Intermediate Algebra
, 5e – Slide #18 Section 2.5
Parallel and Perpendicular Lines
148 EXAMPLE Write an equation of the line passing through (2,-4) and parallel to the line whose equation is
y
= -3
x
+ 5.
SOLUTION Since the line I want to represent is parallel to the given line, they have the same slope. Therefore the slope of the new line is also
m
= -3. Therefore, the equation of the new line is:
y
– (-4) = -3(
x
–2)
y
+ 4 = -3(
x
–2)
y
+ 4 = -3
x
+ 6
y
= -3
x
+ 2 Substitute the given values Simplify Distribute Subtract 4 from both sides Blitzer,
Intermediate Algebra
, 5e – Slide #19 Section 2.5
Parallel and Perpendicular Lines
149-150 EXAMPLE Write an equation of the line passing through (2,-4) and perpendicular to the line whose equation is
y
= -3
x
+ 5.
SOLUTION The slope of the given equation is
m
1 3 1 3 1 = -3. Therefore, the slope slope
m
follows: Blitzer,
Intermediate Algebra
, 5e – Slide #20 Section 2.5
Parallel and Perpendicular Lines
CONTINUED
y
y
1
m
x
x
1
y
y
4 1 3 1
x
x
3 2 2
y
4
y
1 3 1 3
x
2
x
3 2 3 4
y
1 3
x
2 3 4 1 3 3
y
1 3
x
2 3 12 3
y
1 3
x
14 3
m
1 = and the point (2,-4), Substitute the given values Simplify Distribute Subtract 4 from both sides Common Denominators Common Denominators Simplify Blitzer,
Intermediate Algebra
, 5e – Slide #21 Section 2.5
Parallel and Perpendicular Lines
149 -150 Do Check Point 4 on page 149 Find the equation that passes through (-2, 5) and is parallel to the line y = 3x+1 Blitzer,
Intermediate Algebra
, 5e – Slide #22 Section 2.5
Parallel and Perpendicular Lines
149 -150 Do Check Point 5 on page 150 Find the slope and equation of a line that passes through (-2, -6) and is perpendicular to x+3y=12 Standard format must be changed Blitzer,
Intermediate Algebra
, 5e – Slide #23 Section 2.5
DONE
Parallel and Perpendicular Lines
One line is perpendicular to another line if its slope is the negative reciprocal of the slope of the other line.
The following lines are perpendicular: y = 2x + 6 and y = -(1/2)x – 4 are perpendicular.
y = -4x +5 and y = (1/4)x + 3 are perpendicular.
Blitzer,
Intermediate Algebra
, 5e – Slide #25 Section 2.5
Parallel and Perpendicular Lines
Two lines are parallel if they have the same slope.
The following lines are parallel: y = 2x + 6 and y = 2x – 4 are parallel.
y = -4x +5 and y = -4x + 3 are parallel.
Blitzer,
Intermediate Algebra
, 5e – Slide #26 Section 2.5
Example Modeling female officers In 1995, there were 690 female officers in the Marine Corps, and by 2010 this number had increased to about 1110. Refer to graph in Figure 3.48 on page 214.
a) The slope of the line passing through (1995, 690) and (2010,1110) is
m
1110 2010 690 1995 28 b) The number of female officers increased, on average by about 28 officers per year.
y
y
1
m
x
x
1
y
690 28
x
1995 c) Estimate how many female officers there were in 2006.
y y
690 28 28 2006 690 1995 998
y OR
y
28
x
28 ( 2006 ) 55170 55170
y
56168 55170 998 Write the slope-intercept form of the of the line shown in the graph.
// (2010, 1110) (1995, 690)
y
690
y
28
x
28
x
55860 55170