Warm-Up How would you describe the roof at the right?

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?


The slope or pitch of a roof is defined as the number of vertical inches of rise for every 12 inches of horizontal run.


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.

http://www.mathopenref.com/coordslope.html

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.


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.


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


Graph each of the following: In your notebook

1. x = 1

2. y = −4


A line has a slope of −3 and a y-intercept of (0, 5). Write the equation of the line.

Y = -3x + 5


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).


Find the equation of the line that contains the points (−2, 5) and (1, 2).

y=-x +3


Write the equation of the line shown in the graph.

y= -1/3x + 1/2


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


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

Warm-Up How would you describe the roof at the right?

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