4.2 Linear Graphs Homework: EOO 1-81
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A.
Horizontal and Vertical Lines
·
The slope of a horizontal line is 0.
y = mx + b
y = (0)x +
b
y
= b
·
The equation of a horizontal line is y = b or f(x) = b, where b is a
real number.
Example: f(x) = -2
The y-intercept is (0, -2). The slope is 0. The graph is a horizontal line
through (0, -2).
Example: y = 4
The y-intercept is (0, 4). The slope is 0. The graph is a horizontal line
through (0, 4).
Using any two points on the horizontal line graphed above, we can show by
calculating the slope ratio that the slope is indeed 0. For every x-value, the
corresponding y-value is 4. For example, two points on the line are (0, 4) and
(3, 4). The slope ratio is
·
The slope of a vertical line is undefined.
·
The equation of a vertical line is x = c, where c is a real number.
·
The graph is a vertical line through the x-intercept (c, 0). A vertical
line is not a function.
Example:
x = 5
The slope is undefined. The graph is a vertical line through the x-intercept
(5, 0).
Using any two points on the line, for example, (5, 3) and (5, -2), we can
calculate the slope ratio to show that the slope of a vertical line is
undefined.
Example: x = -1
The slope is undefined. The graph is a vertical line through the x-intercept
(-1, 0).
Example: Find the slope of each line.
a) 5y – 12 = 3x
b) –12 = 4x – 7
c) 19 = -6y
d) 2x – 3y = 18
To find the slope, get the equation in the form y = mx + b, slope is m,
or y = b, slope is 0, or
x = c, slope is undefined.
a)
5y – 12 = 3x
5y = 3x +
12
The slope is 
.
b) -12 = 4x – 7
-5 = 4x
The slope is undefined. (Equation is form x = c, a vertical line).
c) 19 = -6y
The slope is 0. (Equation is form y = b, a horizontal line).
d) 2x – 3y = 18
-3y = -2x + 18
The slope is 
.
B. Intercepts
To find the y-intercept, let x = 0 and solve for y.
To find the x-intercept, let y = 0 and solve for x
|
x |
y |
|
|
0 |
|
y-intercept |
|
|
0 |
x-intercept |
Example: Find the intercepts, then
graph the line.
3x – 5y = 15 y-int: 3(0) – 5y =
15 x-int: 3x
– 5(0) = 15
y = -3 x = 5
|
x |
y |
|
|
0 |
-3 |
y-intercept |
|
5 |
0 |
x-intercept |
Graph:
C.
Parallel Lines
Parallel lines (non-vertical) have the same slope.
Without
graphing, determine whether a pair of lines are parallel. Get each equation in
the
y = mx + b form to find the slope of each line. If the slopes are the same,
then the lines are parallel.
Example:
4x – 7 = y
y – 4x = 8
First
equation:
y = 4x – 7, slope is 4
Second equation: y = 4x
+ 8, slope is 4
The slopes are the same, so the lines are parallel.
D. Perpendicular Lines
The product of the slopes of perpendicular lines is –1. (One
exception is that a horizontal line and a vertical line are perpendicular).
Without graphing, determine whether a pair of lines are parallel. Get each
equation in the
y = mx + b form to find the slope of each line. If the product of the
slopes is –1, then the lines are perpendicular.
Example:
3y = 2x – 6
3x + 2y = 8
First
equation:
,
slope is 2/3.
Second equaton: 2y
= -3x + 8
, slope is –3/2.
Multiply
the slopes 
The product of the slopes is –1, so the lines are perpendicular.
Example: Find an equation for a
linear function that is parallel to the given line and that has the given
y-intercept. y = -5x + 2;
(0, -9)
If the line is parallel, then it has the same slope as the given line, m = -5.
The y-intercept is given (0, -9), so b = -9.
Example: Find an equation for a
linear function that is perpendicular to the given line and that has the given
y-intercept.
If the line is perpendicular, then the slope is the negative reciprocal of the
slope in the given line,
.
The y-intercept is given (0, 4), so b = 4.
