More
About
Lines
Point / Slope Form
If a line is written in the form
y – y1 = m(x – x1)
then
m = slope
(x1,y1) is a point on the line
So …
y – 5 = ¾(x + 2)

contains the point (-2,5)

has a slope of ¾
Find an equation of a line
through (4,-3) with a slope of 8
Find an equation of a line
through (4,-3) with a slope of 8
x 1 y1
m
y – y1 = m(x – x1)
Find an equation of a line
through (4,-3) with a slope of 8
x 1 y1
m
y – y1 = m(x – x1)
y + 4 = 8(x – 4)
Find an equations of these
lines:
through (5,7) with slope = -½
through (4,0) with slope = 9
Find an equations of these
lines:
through (5,7) with slope = -½
y – 7 = -½(x – 5)
through (4,0) with slope = 9
y = 9(x – 4)
Find the equation of this line
Find the equation of this line
through (-3,2) m = -½
y – 2 = -½(x + 3)
Find the equation of a line
through (4,7) and (-1,10).
FIRST find the slope
10 – 7
3
m = ---------- = -----1 – 4
-5
Find the equation of a line
through (4,7) and (-1,10).
FIRST find the slope
10 – 7
3
m = ---------- = -----1 – 4
-5
Use EITHER point for (x1,y1)
3
y – 7 = - /5(x – 4)
Find the equation of a line
through (4,7) and (-1,10).
FIRST find the slope
10 – 7
3
m = ---------- = -----1 – 4
-5
Use EITHER point for (x1,y1)
3
y – 7 = - /5(x – 4)
3
or y – 10 = - /5(x + 1)
Find an equation for this line:
Find an equation for this line:
Slope = -3
Find an equation for this line:
y – 4 = -3(x – 2)
Find an equation for this line:
… or y + 5 = -3(x – 5)
These are different names for
the same line.
y – 4 = -3(x – 2)
y – 4 = -3x + 6
y = -3x + 10
y-intercept
is 10
either way
y + 5 = -3(x – 5)
y + 5 = -3x + 15
y = -3x + 10
“Standard” or General Form of
a Line
It’s fairly common to write
lines in the form Ax + By = C.
When this happens, the
easiest way to graph them is
to use the intercepts (the
points where x and y are 0).
Find the intercepts of
4x + 6y = 12
Find the intercepts of
4x + 6y = 12
Find (0,___) and (___,0)
Find the intercepts of
4x + 6y = 12
Find (0,___) and (___,0)
If x = 0, then 6y = 12
 y=2
Find the intercepts of
4x + 6y = 12
Find (0,___) and (___,0)
If x = 0, then 6y = 12
 y=2
If y = 0, then 4x = 12
 x=3
(0,2) and (3,0) are intercepts.
Find the intercepts of
4x + 6y = 12
Find (0,___) and (___,0)
If x = 0, then 6y = 12
 y=2
If y = 0, then 4x = 12
 x=3
(0,2) and (3,0) are intercepts.
Parallel Lines



In the same plane
Never intersect
Same distance apart
Most important
 Have the SAME SLOPE
Perpendicular Lines

Meet to form a right angle

Slopes are OPPOSITE
RECIPROCALS

Meet to form a right angle

Slopes are OPPOSITE
RECIPROCALS
-2 and ½
3/
5
and - 5/3
Find the equation of a line through
(5,-7) that is parallel to y = 4/9x + 1
Find the equation of a line through
(5,-7) that is parallel to y = 4/9x + 1
y + 7 = 4/9(x – 5)
Find the equation of a line
perpendicular to y – 3 = ¾(x – 2),
with a y-intercept at 7.
Find the equation of a line
perpendicular to y – 3 = ¾(x – 2),
with a y-intercept at 7.
y = -4/3x + 7
Are these lines parallel,
perpendicular, or neither?
y = 5x – 2 and y = -5x + 3
y = 2x + 7 and y = 2x + 4
y = ¼x + 2 and y = -4x + 2
Are these lines parallel,
perpendicular, or neither?
y = 5x – 2 and y = -5x + 3
Neither
y = 2x + 7 and y = 2x + 4
Parallel
y = ¼x + 2 and y = -4x + 2
Perpendicular
Direct Variation
 Something changes at a
constant rate
 One thing is a multiple of
another
Direct variations can always
be written in the form y = kx
(a line with a y-intercept of 0)

k is called the constant of
variation (same as the
slope of the line)
The distance from lightning
varies directly as the time it
takes to hear thunder. If you
hear thunder 10 seconds after
a lightning flash, the lightning
is 2 seconds away.


Write a direct variation.
How long would it take to
hear thunder from a flash
25 miles away?
Write a direct variation.
y = kx
2 = k  10
1/ = k
5
So … y = 1/5 x
How long would it take to hear
thunder from a flash 25 miles
away?
1/ x
5
1/  25
5
y=
y=
y = 5 seconds
Weight on Mars varies directly
as weight on Earth. Manuel
Uribe, the heaviest man who
ever lived on Earth
weighed 1,230
pounds. On Mars,
Manuel would
weigh 467 pounds.
Write a direct variation.
How much would you weigh on
Mars?
If an alien weighed
13 pounds on her
home planet of Mars,
how much would
the alien weigh
on earth?
Write a direct variation.
Earth = 1230  Mars = 467
y = kx
467 = k  1230
.38 = k
So … y = .38x
If an alien weighed 13 pounds on
Mars, how much would the alien
weigh on earth?
13 = .38x
about 32 pounds
Graphs of Absolute Value
Functions
 Always a
V-shape

Part of a line
and a
reflection
of that line
We have already graphed
y=|x|
Now consider these graphs
y=|x|+3
y=|x|–2
What would y = | x | – 6 look
like?
What would y = | x | – 6 look
like?
Remember y = | x |
Now consider these graphs
y = 4| x |
y=½|x|
What would y = 2| x | look
like?
What would y = 2| x | look
like?
What would y = -| x | look like?
What would y = -| x | look like?
Remember y = | x |
Now consider these graphs
y=|x+3|
y=|x–4|
What would y = | x + 2 | look
like?
What about y = | x – 7 | ?
What would y = | x + 2 | look
like?
What about y = | x – 7 | ?
Now put it all together.
What would the graph of
y = -| x + 5 | – 1 look like?
Now put it all together.
What would the graph of
y = -| x + 5 | – 1 look like?
 left 5
 down 1
 upside down
Now put it all together.
What would the graph of
y = -| x + 5 | – 1 look like?
 left 5
 down 1
 upside down
What would the graph of
y = ¼| x – 3 | + 4 look like?
What would the graph of
y = ¼| x – 3 | + 4 look like?
 right 3
 up 4
 slope = ¼
What would the graph of
y = ¼| x – 3 | + 4 look like?
 right 3
 up 4
 slope = ¼
Identify these functions.
y=|x–1|
y=|x–2|+3
y = -| x + 6 | – 2