lines with zero slope and undefined slope example use slope-intercept form to graph f (x) = 2...
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Lines with Zero Slope and Undefined Slope
Example Use slope-intercept form to graph f (x) = 2
SolutionWriting in slope-intercept form: f (x) = 0 • x + 2. No matter what number we choose for x, we find that y must equal 2.
Choose any number for x
y must always be 2
x y (x, y)
0 2 (0, 2)
4 2 (4, 2)
4 2 (4 , 2)
f (x) = 2
Example continued y = f (x) = 2
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).
x y (x, y)
2 4 (2, 4)
2 1 (2, 1)
2 4 (2, 4)
x must be 2
Example Graph x = 2
SolutionWe regard the equation x = 2 as x + 0 • y = 2. We make up a table with all 2 in the x-column.
Any number can be used for y
x = 2
Example continued x = 2
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).
Example
Find the slope of each given line. a. 4y + 3 = 15 b. 5x = 20
Solutiona. 4y + 3 = 15 (Solve for y)
4y = 12 y = 3
Horizontal line, the slope of the line is 0. b. 5x = 20 x = 4 Vertical line, the slope is undefined.
Graphing Using Intercepts
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, (a, 0) where a is a constant.
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, (0, b) where b is a constant.
x-intercepts
(3, 0)(-1, 0)
y-intercept
(0, -3)
Example Graph 5x + 2y = 10 using intercepts.
SolutionTo 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)
Go to your y-editor and enter the equation.
Select menu 1
Enter 0
So, the y-intercept is (0, -4).
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Select 2 from the menu
Enter 3
Press
Enter 8
Press
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So, the x-intercept is (6, 0).
Press
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5Change window size to [ 10,5] [ 5,20]
Press
The three amigos use the notation
[ 10,5, 5,20] for window size.
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.
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, 0).
Both lines have slope 3/2 and different y-intercepts, the graphs are parallel.
33
2y x
33
2y x
3 52 2
y x
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.
2 1
3 3y x
2
3
1
3y x
3 2 1x y
2 1 3y x
1 3
2 2
xy
1
2
3
2y x
The slope of the line is 3/2. The lines are perpendicular if the product of their slopes is 1.
2 31
3 2
The lines are perpendicular.
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 (a) We begin by determining the slope of the line.
4 5 3x y
5 4 3y x 4 3
5 5y x The slope is
4.
5
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
45 .
45
5y x
continued
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
545 4or .
55
4y x
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They don’t appear to bePerpendicular.
Press
Select 5 from the menu
That’s better!
Recognizing Linear Equations
A linear equation may appear in different forms, but all linear equations can be written in standard form Ax + By = C.
Example
Determine whether each of the following equations is linear: a) y = 4x + 2 b) y = x2 + 3
Solution Attempt to write each equation in standard form. a) y = 4x + 2 4x + y = 2 Adding 4x to both sides
b) y = x2 + 3 x2 + y = 3 Adding x2 to both sides The equation is not linear since it has an x2 term.