geometry unit 3.7
TRANSCRIPT
UNIT 3.7 EQUATIONS OF LINES IN THE COORDINATE PLANE
Slope (m)
The ratio of its vertical rise to its horizontal run.
Steepness
Slope = m =Vertical rise
Horizontal run
Find the slopes.
28m
28m
-8
2
42
4m 2
2
4
Slope (continued)
12
12
xxyy
m
of a line containing two points with coordinates (x1, y1) and (x2, y2) is given by the formula
Slopes
12
12
xxyy
m
3155
All horizontal lines have a 0
slope
All vertical lineshave an
undefined slope
40 0
12
12
xxyy
m
66
34
07 undefined
Positive Slopes
Rise (upward) as you move left to right
Line slopes up from left
to right
y
x
Negative Slope
Fall (downward) as you move left to right
Line slopes down from left to right
y
x
Find the slope using the slope formula.
12
12
xxyy
m
87
62
25
87
12
12
xx
yym
04
10
41
41
Rate of Change
Describes how a quantity is changing over time.
xy
The slope of a line can be used to determine the Rate of Change
Change in quantity (y) Change in time (x)
Recreation: For one manufacturer of camping equipment,
between 1990 and 2000 annual sales increased by $7.4 million
per year. In 2000, the total sales were $85.9 million. If the
sales increase at the same rate, what will be the total sales in
2010?12
12
xxyy
m
14.7
20002010
9.852
y
10
9.85
1
4.72y
+85.9 +85.9
159.9 mill. = y2
74.0 = y2 – 85.9
7.4(10) = y2 – 85.9
Forms of Linear Equations
Slope-Intercept Form -
y = mx + b
slope y-intercept
Point-Slope Form -
y – y1 = m(x – x1)
slope x-coordinatey-coordinate
Graph
13
2 xy
13
2 xy
1.) The equation is in slope-intercept form y = mx + b
32
The slope is
y-intercept (0, 1)
2.) Plot the point (0, 1)
32
3.) Use the slope , from
the point (0, 1) go up 2,right 3
Graph
13
2 xy
13 xy
1.) The equation is in slope-intercept form y = mx + b
The slope is 3
y-intercept (0, 1)
2.) Plot the point (0, 1)
3.) Use the slope 3, from the point (0, 1) go up 3,right 1
Graph y - 3 = -2(x + 3)
1.) The equation is in point-slope form y – y1 = m(x – x1)
The slope is -2
Point on line (-3, 3)
2.) Plot the point (-3, 3)
3.) Use the slope -2, from the point (-3, 3) go down 2, right 1
Graph )4(3
12 xy
Point on line (4, 2)
2.) Plot the point (4, 2)
31The slope is
3.) Use the slope , from
the point (4, 2) go down 1, right 3
31
1.) The equation is in point-slope form y – y1 = m(x – x1)
Writing Equations of Linear Lines
If we know the slope and at least one point
If we have the slope and y-intercept, use the slope-intercept form; y = mx + b
If we have the slope and a point, use the point-slope form; y – y1 = m(x – x1)
Write the equation of the line
What is an equation of the line with slope 3 and y-intercept -5?
Start with the slope-intercept form of the equation
y = mx + by = 3x + (-5) Substitute 3 for m, and -5
for b
Simplifyy = 3x - 5
Write the equation of the line
What is an equation of the line through point (-1, 5) with slope 2?
Start with the point-slope form of the equation
y – y1 = m(x – x1)
y – 5 = 2(x - (-1)) Substitute 2 for m, and -1 in for x1 and 5 in for y1
Simplifyy – 5 = 2(x + 1)
Write the equation of the line
21
What is an equation of the line with slope and y-intercept 2?
Start with the slope-intercept form of the equation
y = mx + b
Substitute for m, and 2 for b 2
1y = x + 221
Write the equation of the line
What is an equation of the line through point (-1, 4) with slope -3?
Start with the point-slope form of the equation
y – y1 = m(x – x1)
y – 4 = -3(x - (-1)) Substitute -3 for m, and -1 in for x1 and 4 in for y1
Simplifyy – 4 = -3(x + 1)
Writing Equations of Linear Lines
If we know two points on the line Find the slope using the formula Using the point-slope formula Plug in one of the two points Plug in the slope for m
Write the equation of the line
What is an equation of the line through point (-2, -1) and point (3, 5)?
12
12
xxyy
m Find the slope
y + 1 = (x + 2) or56
y - 3 = (x - 5)56
32
51
56
56
Start with the point-slope form of the equation
y – y1 = m(x – x1) Plug in the slope and one of the two points
Writing Equations Horizontal and Vertical Lines
We don’t need a slope All points on a horizontal line have the
same y-coordinate; so the equation is y = y1.
All points on a vertical line have the same x-coordinate; so the equation is x = x1.
Where (x1, y1)
Write the equation of the line
What are the equations for the horizontal and vertical lines through (2, 4)?
The horizontal is y = y1
y = 4 Substitute 4 for y1
The vertical is x = x1
x = 2 Substitute 2 for x1
Write the equation of the line
What are the equations for the horizontal and vertical lines through (4, -3)?
The horizontal is y = y1
y = -3 Substitute -3 for y1
The vertical is x = x1
x = 4 Substitute 4 for x1
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