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2.3
Another Look at Linear Graphs
■ Graphing Horizontal Lines and Vertical Lines
■ Graphing Using Intercepts
■ Parallel and Perpendicular Lines
■ Recognizing Linear Equations
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3. Graphing Horizontal Lines and Vertical Ines
Slope of a Horizontal Line
The slope of a horizontal line is 0.
The graph of any function of the form f(x) = b or y = b is a horizontal line that crosses the y-axis as (0, b).
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Example Use slope-intercept form to graph f(x) = 2
Solution
Writing in slope-intercept form: f(x) = 0 • x + 2. No matter what number we choose for x, we find that y must equal 2.
y = 2
Choose any number for x.
y must be 2. x y
(x, y) 2
(0, 2) 4
(4, 2) 4
(4 , 2)
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Example continued y = 2
Solution
When we plot the ordered pairs (0, 2), (4, 2) and (4, 2) and connect the points, we obtain a horizontal line.
Any ordered pair of the form (x, 2) is a solution, so the line is parallel to the x-axis with y-intercept (0, 2).
y = 2
(4, 2)
(0, 2)
(4, 2)
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Slope of a Vertical Line
The slope of a vertical line is undefined.
The graph of any equation of the form x = a is a vertical line that crosses the x-axis as (a, 0).
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Example Graph x = 2
Solution
We regard the equation x = 2 as x + 0 • y = 2. We make up a table with all 2 in the x-column.
x = 2
x must be 2. x y
(x, y) 2 4
(2, 4) 1
(2, 1) 4
(2, 4)
Any number can be used for y.
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Example continued x = 2
Solution
When we plot the ordered pairs (2, 4), (2, 1), and (2, 4) and connect them, we obtain a vertical line.
Any ordered pair of the form (2, y) is a solution. The line is parallel to the y-axis with x-intercept (2, 0).
x = 2
(2, 4)
(2, 4)
(2, 1)
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Example
Find the slope of each given line. If the slope is undefined state this.
a. 4y + 3 = 15 b. 5x = 20
Solution
a. 4y + 3 = 15 (Solve for y)
4y = 12
y = 3
The slope of the line is 0.
b. 5x = 20
x = 4
The slope is undefined.
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10. Graphing Using Intercepts
The point at which the graph crosses the y-axis is called the y-intercept. The x-coordinate of a
y-intercept is always 0.
The point at which the graph crosses the x-axis is called the x-intercept. The y-coordinate of a
x-intercept is always 0.
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To Determine Intercepts
The x-intercept is (a, 0). To find a, let y = 0 and solve the original equation for x.
The y-intercept is (0, b). To find b, let x = 0 and solve the original equation for y.
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Example Graph 5x + 2y = 10 using intercepts.
Solution
To find the y-intercept we let x = 0 and solve for y.
5(0) + 2y = 10
2y = 10
y = 5 (0, 5)
To find the x-intercept we let y = 0 and solve for x.
5x + 2(0) = 10
5x = 10
x = 2 (2, 0)
5x + 2y = 10
x-intercept (2, 0)
y-intercept (0, 5)
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Example Graph 3x  4y = 8 using intercepts.
Solution
To find the y-intercept, we let x = 0. This amounts to ignoring the x-term and then solving.
4y = 8
y = 2
The y-intercept is (0, 2).
To find the x-intercept, we let y = 0. This amounts to ignoring the y-term and then solving.
3x = 8
x = 8/3
The x-intercept is (8/3, 0).
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Example continued Graph 3x  4y = 8 using intercepts.
Solution
Find a third point. If we let x = 4, then
3 • 4  4y = 8
12  4y = 8
4y = 4
y = 1
We plot (0, 2); (8/3, 0); and (4, 1).
x-intercept
y-intercept
3x  4y = 8
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15. Parallel and Perpendicular Lines
Two lines are parallel if they lie in the same plane and do not intersect no matter how far they are extended. If two lines are vertical, they are parallel.
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16. Slope and Parallel Lines
Two lines with different y-intercepts are parallel if they have the same slope. Also, two vertical lines are parallel.
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Example Determine whether the graphs of
and 3x  2y = 5 are parallel.
Solution When two lines have the same slope but different y-intercepts they are parallel.
The line has slope 3/2 and y-intercept (0, 3).
Rewrite 3x  2y = 5 in slope-intercept form:
3x  2y = 5
2y = 3x  5
The slope is 3/2 and the y-intercept is 5/2.
Both lines have slope 3/2 and different y-intercepts, the graphs are parallel.
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Slope and Perpendicular Lines
Two lines are perpendicular if the product of their slopes is 1 or if one line is vertical and the other is horizontal.
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Example Determine whether the graphs of 3x  2y = 1 and are perpendicular.
Solution
First, we find the slope of each line.
The slope is – 2/3.
Rewrite the other line in slope-intercept form.
The slope of the line is 3/2. The lines are perpendicular if the product of their slopes is 1.
The lines are perpendicular.
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Example
Write a slope-intercept equation for the line whose graph is described.
a) Parallel to the graph of 4x  5y = 3, with y-intercept (0, 5).
b) Perpendicular to the graph of 4x  5y = 3,
with y-intercept (0, 5).
Solution
We begin by determining the slope of the line.
The slope is
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continued
a) A line parallel to the graph of 4x  5y = 3 has a slope of Since the y-intercept is (0, 5), the slope-intercept equation is
b) A line perpendicular to the graph of 4x  5y = 3 has a slope that is the negative reciprocal of Since the y-intercept is (0, 5), the slope-intercept equation is
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