9/2/2013
Outline: Chapter 1 - Lines
Review
Graphing lines - using tables / points
Slope
- calculating via
- points
- equation
- table
- meaning
- parallel and perpendicular lines
- useful for graphing
Equations of lines
- from two points or table (approximate if decimals)
- from slope & point (parallel or perpendicular line through a given point)
Graphing Lines
- using y = mx + b
- finding slope from graph
slope =
vertical distance
horizontal distance
grade = 100%⋅ slope
Finding the Slope of a Line
Investigating Slope of a Horizontal Line
Fi n d i n g a Li n e ’s S lop e
H o r i z o n t a l a n d Ve r t i c a l L i n e s
Example 1
Example 2
Find the slope of the line that contains the points
(2, 3) and (5, 1).
Find the slope of the line that
contains the points (2, 3) and
(6, 3).
Graph
The slope of the horizontal line is ____, ____ steepness.
Copyright © 2011 Pearson Education, Inc.
Copyright © 2011 Pearson Education, Inc.
Copyright © 2011 Pearson Education, Inc.
Section 1.3
Lehmann, Intermediate Algebra, 4ed
Slide 12
Copyright © 2011 Pearson Education, Inc.
Section 1.3
Lehmann, Intermediate Algebra, 4ed
Slide 13
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9/2/2013
Investigating the slope of a Vertical Line
Property
H o r i z o n t a l a n d Ve r t i c a l L i n e s
H o r i z o n t a l a n d Ve r t i c a l L i n e s
Example 3
Property
• A __________line has slope of zero (_____ figure).
Find the slope of the line that
contains the points (4, 2) and
(4, 5).
• A ________ line has undefined slope (_____ figure).
The slope of the vertical line is __________.
Section 1.3
Copyright © 2011 Pearson Education, Inc.
Copyright © 2011 Pearson Education, Inc.
Copyright © 2011 Pearson Education, Inc.
Slide 14
Lehmann, Intermediate Algebra, 4ed
Copyright © 2011 Pearson Education, Inc.
Section 1.3
Slide 15
Lehmann, Intermediate Algebra, 4ed
Finding Slopes of Parallel Lines
Finding Slopes of Parallel Lines
Par allel an d Per p en d icu lar Lin es
Par allel an d Per p en d icu lar Lin es
Definition
Solution
Two lines are called parallel
if they do not intersect.
• Nonvertical parallel lines
have ______ slope, since they
have the same steepness
Example 4
Property
If lines l1 and l2 are nonvertical parallel lines on the
same coordinate system, then the slopes of the lines
are equal:
___ = ____
Also, if two distinct lines have equal slope, then the
lines are __________.
Find the slopes of the lines l1
and l2 sketch to the right.
Copyright © 2011 Pearson Education, Inc.
Copyright © 2011 Pearson Education, Inc.
Section 1.3
Lehmann, Intermediate Algebra, 4ed
Slide 16
Copyright © 2011 Pearson Education, Inc.
Section 1.3
Lehmann, Intermediate Algebra, 4ed
Slide 17
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9/2/2013
Property
Finding Slopes of Perpendicular Lines
Par allel an d Per p en d icu lar Lin es
Par allel an d Per p en d icu lar Lin es
Definition
Example 5
Two lines are perpendicular is
they intercepts at a 90o angle.
Find the slopes of the
perpendicular lines l1 and l2.
Solution
Copyright © 2011 Pearson Education, Inc.
Copyright © 2011 Pearson Education, Inc.
Copyright © 2011 Pearson Education, Inc.
Section 1.3
Slide 18
Lehmann, Intermediate Algebra, 4ed
Property
Copyright © 2011 Pearson Education, Inc.
Section 1.3
Lehmann, Intermediate Algebra, 4ed
Slide 19
1.5 Defining Point-Slope Form
Method :
Par allel an d Per p en d icu lar Lin es
Property
If lines l1 and l2 are nonvertical perpendicular lines,
then the slope of one line is the ________ of the
_____________ of the slope of the other line:
Also, if the slope of one line is the opposite of the
reciprocal of another line’s slope, then the lines are
_______________.
One method to find a linear equation of a line, that
ALWAYS works.
Suppose that a non-vertical line has:
• Slope = m
• coordinate = (x1, y1)
• (x, y) represents a different point on the line
So, the slope is:
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Using Point-Slope
y − y1
=m
x − x1
Copyright © 2011 Pearson Education, Inc.
Copyright © 2011 Pearson Education, Inc.
Section 1.3
Lehmann, Intermediate Algebra, 4ed
Slide 20
Section 1.5
Lehmann, Intermediate Algebra, 4ed
Slide 21
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9/2/2013
Defining Point-Slope Form
Method :
But I like y = mx + b……
Using Point-Slope
Why
Given the slope, multiple both sides by x – x1 gives
y − y1
·(x – x1) = m (x – x1)
x − x1
y – y1 = m (x – x1)
We say that this linear equation is in point-slope
form.
Definition
If a nonvertical line has slope m and contains the point
(x1, y1), then an equation of the line is
y – y1 = m (x – x1)
Copyright © 2011 Pearson Education, Inc.
Section 1.5
Lehmann, Intermediate Algebra, 4ed
Slide 22
Using Point-Slope Form to Find an Equation of a Line
Method :
y – y1 = m (x – x1) ?
• Can ALWAYS use this equation for line!
• Method is SIMPLE!!!
• Just need ONE point (if given more, just pick
ONE)
• Need slope (if not given, then calculate)
(This is WAY more useful than y = mx+b because
you can use ANY point, whereas y =mx + b
REQUIRES the y-intercept, which you will
rarely know. If you have the y-intercept the
above equation still works. Cool!)
Using Point-Slope Form to Find an Equation of a Line
Using Point-Slope
Method :
Using Point-Slope
Example 1
Example 2
A line has slope m = 2 and contains the point (3, –8).
Find the equation of the line
Use point-slope form to find an equation of the line
that contains the points (–5, 2) and (3, –1). Then
write in slope-intercept form.
Solution
Solution
Copyright © 2011 Pearson Education, Inc.
Copyright © 2011 Pearson Education, Inc.
Section 1.5
Lehmann, Intermediate Algebra, 4ed
Slide 24
Section 1.5
Lehmann, Intermediate Algebra, 4ed
Slide 25
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Slope / Equation from a table
Slope / Equation
Slope Property
Slope Property
Example 3
For the following sets, is there a line that passes
through them? If so, find the slope of that line.
Solution
• Value of x increases by ___.
• Value of y changes by ___.
• The slope is ____.
Copyright © 2011 Pearson Education, Inc.
Copyright © 2011 Pearson Education, Inc.
Copyright © 2011 Pearson Education, Inc.
Section 1.4
Slide 26
Lehmann, Intermediate Algebra, 4ed
Copyright © 2011 Pearson Education, Inc.
Section 1.4
Lehmann, Intermediate Algebra, 4ed
Slide 27
Equation of line from chart
Example: Find equation of line from set 1.
CA – Find equation of line from set 2 & set 3,
when possible.
Copyright © 2011 Pearson Education, Inc.
Lehmann, Intermediate Algebra, 4ed
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