Download The Point-Slope Form of the Equation of a Line I. Point

Math 152 — Rodriguez
The Point-Slope Form of the Equation of a Line
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I. Point-Slope Form
A. Linear equations we have seen so far:
1. standard form:Ax +By=C
2. slope-intercept form:
3. horizontal line:
4. vertical line:
A, B, and C real numbers
y = mx + b, where m slope; y-intercept (0,b)
y=#
x=#
B. Let (x1, y1) be a specific (fixed) point on the line and (x,y) be a random (general) point
on the line. If we plug these into the slope formula and re-arrange it we get:
m=
y − y1
x − x1
The point-slope form of the equation is
where m is the slope and (x1, y1) is point on the line.
II. Using the Point−Slope Form to Write the Equation of a Line
A. Given a point and slope: plug info into the equation.
Example: slope= 4, passing through (2, 5)
y
−
y
−
y1 = m (x
=
(x
−
x 1)
−
)
You will be asked to write the final answer in slope-intercept form so…
If you are told to use function notation then:
B. Given two points: find the slope, pick one of the points, plug into equation.
Example: passing through (3,8) and (5,4)
Steps:
1. Find m =
2. Pick a point:
3. Plug: y
−
y1 = m (x
−
x 1)
Again, want final answer in slope-intercept form so you’d keep going.
C. Instructions you will see for hw:
Write the point-slope form of the line’s equation satisfying the conditions. Then use the
point-slope form of the equation to write the slope-intercept form of the equation in
function notation.
1) slope=
3)
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− 2 , passing through (−1, 5)
3
2) slope= −3, passing through (9, 5)
passing through (4, 2) and (6,−2)
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III.
Parallel Lines
A. Parallel lines: two lines that ‘grow’ at the same rate and so never intersect.
B. Parallel lines have the same slope (or are vertical lines).
Example 1: Write an equation for line L in point-slope form and slope-intercept form.
y =2x +
3
L
(6,4)
Example 2: Find the point-slope form and slope-intercept form of the equation for the line
passing through (−3,4) and parallel to the line whose equation is y = −2x+5.
IV.
Perpendicular Lines
A. Perpendicular lines: lines that intersect at a 90° angle.
C. Perpendicular lines have slopes whose product is −1.
D. The slope of one line is the negative reciprocal of the slope of the other line (or one
line is horizontal and the other is vertical).
E. Line L and M are perpendicular
Slope line L Slope line M
−
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line L
5
y = 2x+3
2
3
3x − 4y = 8
Slope line L Slope line M
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F. Example: Find the point-slope form and slope-intercept form of the equation for the line
passing through (−3,4) and perpendicular to the line whose equation is y = −2x+5.
V. Parallel and Perpendicular Lines
Examples: Use the given conditions to write an equation for each line in point-slope and
slope-intercept form.
1) Passing through (8, −3) and parallel to line whose equation is 3x + 4y = 6
2) Passing through (3, 2) and perpendicular to line whose equation is y = 3x+5
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VI. Applications
Same application problems as section 2.4 except now we are not given the y-intercept.
Retail Sales
(billions of $)
1) The bar graph shows retail sales, in billions of dollars, of nonfood pet supplies from
2001 to 2005.
10
8
6
4
2
0
7.2
9
8.5
7.9
7.6
(4, 9)
(1, 7.6)
0
1
2
3
4
5
Years after 2001
a. Let x represent the number of years since 2001. Let y represent the amount spent
on nonfood pet supplies, in billions of dollars. Use the coordinates of the points
shown to find the point-slope form of the equation of the line that models the
amount of money spent on nonfood pet supplies, y, x years after 2001.
b. Write the equation from part (a) in slope-intercept form.
c. If this trend continued, use the equation from part (b) to predict the amount spent
on nonfood dog supplies in 2010.
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Sentenced Inmates
(per 100,000 residents)
2) The following graph is a scatter plot that shows the number of sentenced inmates in
the U.S. per 100,000 residents from 2001 to 2005. Also shown is a line that passes
through or near the data points.
490
485
480
(4, 486)
475
470
(2, 476)
465
0
1
2
3
4
5
Years after 2000
a) Use the two points whose coordinates are shown by the voice balloons to find the
point-slope form of the equation that models the number of inmates per 100,000
residents, y, x years after 2000.
b) Write the equation from part (a) in slope-intercept form. Use function notation.
c) Use the linear function to predict the number of sentenced inmates in the U.S. per
100,000 residents in 2010.
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