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Warm-UpWarm-Up
How would you describe the roof at the right?
Warm-UpWarm-Up
Anything that isn’t completely vertical has a slopeslope. This is a value used to describe its incline or decline.
Warmer-UpperWarmer-Upper
The slope or pitchpitch of a roof is quite a useful measurement. How do you think a contractor would measure the slope or pitch of a roof?
Warmer-UpperWarmer-Upper
The slope or pitch of a roof is defined as the number of vertical inches of rise for every 12 inches of horizontal run.
Warmer-UpperWarmer-Upper
The steeper the roof, the better it looks, and the longer it lasts. But the cost is higher because of the increase in the amount of building materials.
3.4 Find and Use Slopes of Lines3.4 Find and Use Slopes of Lines3.5 Write and Graph Equations of Lines3.5 Write and Graph Equations of Lines
Objectives:
1. To find the slopes of lines
2. To find the slopes of parallel and perpendicular lines
3. To graph and write equations based on the Slope-Intercept Form, Standard Form, or Point-Slope Form of a Line
Investigation 1Investigation 1
Click on the button and use the activity, to discover something about the actual value of the slope of a line. Then complete the table on the next slide.
Slope SummarySlope Summary
Summarize your findings about slope in the table below:
m > 0 m < 0 m = 0 m = undef
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As the absolute value of the slope of a line increases, --?--.the line gets steeper.
Copy and complete in your notebook
Slope of a LineSlope of a Line
The slope of a line (or segment) through P1 and P2 with coordinates (x1,y1) and (x2,y2) where x1x2 is
rise
rise
Example 2Example 2
Find the slope of the line containing the given points. Then describe the line as rising, falling, horizontal, or vertical.
• (6, −9) and (−3, −9)
• (8, 2) and (8, −5)
• (−1, 5) and (3, 3)
• (−2, −2) and (−1, 5)
0 horizontal
undefined vertical
-1/2 falling
7 rising
Example 3Example 3
A line through points (5, -3) and (−4, y) has a slope of −1. Find the value of y.
Parallel and PerpendicularParallel and Perpendicular
Two lines are parallel parallel lines lines iff they have the same slope.
Two lines are perpendicular lines perpendicular lines iff their slopes are negative reciprocals.
Example 4 Example 4
Tell whether the pair of lines are parallel, perpendicular, or neither
1.Line 1: through (−2, 1) and (0, −5)
Line 2: through (0, 1) and (−3, 10)
2.Line 1: through (−2, 2) and (0, −1)
Line 2: through (−4, −1) and (2, 3)
-3 & -3
parallel
-3/2 & 2/3
perpendicular
Example 5Example 5
Line k passes through (0, 3) and (5, 2). Graph the line perpendicular to k that passes through point (1, 2).
Find slope, use it’s negative reciprocal to find slope of new line, then use new slope to plot the 2nd point of new line.
Example 6Example 6
Find the value of y so that the line passing through the points (3, y) and (−5, −6) is perpendicular to the line that passes through the points (−2, −7) and (10, 1).
-18
Example 7Example 7
Find the value of k so that the line through the points (k – 3, k + 2) and (2, 1) is parallel to the line through the points (−1, 1) and (3, 9).
K=2
InterceptsIntercepts
The xx-intercept-intercept of a graph is where it intersects the x-axis.• (aa, 0)
The yy-intercept-intercept of a graph is where it intersects the y-axis.• (0, bb)
6
4
2
-2
-5 5x-intercept
y-intercept
Slope-InterceptSlope-Intercept
Slope-Intercept Form of a Line:Slope-Intercept Form of a Line:If the graph of a line has slope m and a y-intercept of (0, b), then the equation of the line can be written in the form y = mx + b.
Equation of a Horizontal Line
Equation of a Vertical Line
y = b(where b is the y-intercept)
x = a(where a is the x-intercept)
Example 9Example 9
Find the equation of the line with the set of solutions shown in the table.
1.Find slope
2.Plug in x, y and slope into y=mx+b
3.Solve for “b”
4.Write the equation using slope and y-intercept
x 1 3 5 7 9 …
y 5 11 17 23 29 …
3
5=3(1) + b
2
y = 3x + 2
Example 10Example 10
Graph the equation:
32
1 xy
Slope-InterceptSlope-Intercept
To graph an equation in slope-intercept form:
1.Solve for y to put into slope-intercept form.
2.Plot the y-intercept (0, b).
3.Use the slope m to plot a second point.
4.Connect the dots.
Example 11Example 11
Graph the equation:
1032 yx
Standard FormStandard Form
Standard Form of a LineStandard Form of a LineThe standard form of a linear equation is Ax + By = C, where A and B are not both zero.
A, B, and C are usually integers.
Standard FormStandard Form
To graph an equation in standard form:
1.Write equation in standard form.
2.Let x = 0 and solve for y. This is your y-intercept.
3.Let y = 0 and solve for x. This is your x-intercept.
4.Connect the dots.
Example 12Example 12
Without your graphing calculator, graph each of the following: In your notebook
1. y = −x + 2
2. y = (2/5)x + 4
3. f (x) = 1 – 3x
4. 8y = −2x + 20
Example 13Example 13
Graph each of the following: In your notebook
1. x = 1
2. y = −4
Example 14Example 14
A line has a slope of −3 and a y-intercept of (0, 5). Write the equation of the line.
Y = -3x + 5
Example 15Example 15
A line has a slope of ½ and contains the point (8, −9). Write the equation of the line.
HINT: Plug all value into slope-intercept form first
y=1/2x -13
Point-Slope FormPoint-Slope Form
Given the slope and a point on a line, you could easily find the equation using the slope-intercept form.
Alternatively, you could use the point-slope form of a line.
Point-Slope Form of a Line:Point-Slope Form of a Line:A line through (x1, y1) with slope m can be written
in the form y – y1 = m(x – x1).
Example 16Example 16
Find the equation of the line that contains the points (−2, 5) and (1, 2).
y=-x +3
Example 17Example 17
Write the equation of the line shown in the graph.
y= -1/3x + 1/2
Example 18Example 18
Write an equation of the line that passes through the point (−2, 1) and is:
1.Parallel to the line y = −3x + 1
2.Perpendicular to the line y = −3x + 1
y=-3x - 5
y= 1/3x - 5
Example 19Example 19
Find the equation of the perpendicular bisector of the segment with endpoints (-4, 3) and (8, -1).
HINT: find midpoint, then use that point to find formula of new line
y= -3x + 7