Transcript Beginning & Intermediate Algebra, 4ed

§ 3.3
Intercepts
Intercepts
Intercepts of axes (where graph crosses the axes)
Since all points on the x-axis have a y-coordinate
of 0, to find the x-intercept, let y = 0 and solve for
x.
Since all points on the y-axis have an x-coordinate
of 0, to find the y-intercept, let x = 0 and solve for
y.
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Intercepts
Example:
Find the y-intercept of 4 = x – 3y
Let x = 0.
Then 4 = x – 3y becomes
4 = 0 – 3y
Replace x with 0.
4 = – 3y
Simplify the right side.

4
3
=y
Divide both sides by – 3.
So the y-intercept is (0,  43 ).
Martin-Gay, Beginning and Intermediate Algebra, 4ed
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Intercepts
Example:
Find the x-intercept of 4 = x – 3y
Let y = 0.
Then 4 = x – 3y becomes
4 = x – 3(0) Replace y with 0.
4=x
Simplify the right side.
So the x-intercept is (4,0).
Martin-Gay, Beginning and Intermediate Algebra, 4ed
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Graph by Plotting Intercepts
Example:
Graph the linear equation 4 = x – 3y by plotting intercepts.
We previously found that the y-intercept is (0,  43 ) and the
x-intercept is (4, 0).
Along with the intercepts, for the third solution, let y = 1.
4 = x – 3(1) Replace y with 1.
4=x–3
Simplify the right side.
4+3=x
Add 3 to both sides.
7=x
Simplify the left side.
So the third solution is (7, 1).
Continued.
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Graph by Plotting Intercepts
y
Example continued:
4
Plot the two intercepts (0,  3 )
and (4, 0) along with the
third solution (7, 1).
(4, 0)
4
Draw the line
that contains the
three points.
(0,  3 )
Martin-Gay, Beginning and Intermediate Algebra, 4ed
(7, 1)
x
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Graph by Plotting Intercepts
Example:
Graph 2x = y by plotting intercepts
To find the y-intercept, let x = 0
2(0) = y
0 = y, so the y-intercept is (0,0)
To find the x-intercept, let y = 0
2x = 0
x = 0, so the x-intercept is (0,0)
Notice that the x- and y-intercept are the same point.
Continued.
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Graph by Plotting Intercepts
Example continued:
Find two other solutions to the equation.
Let x = 3
2(3) = y
6 = y, so another point is (3, 6)
Let y = 4
2x = 4
x = 2, so another point is (2, 4)
Continued.
Martin-Gay, Beginning and Intermediate Algebra, 4ed
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Graph by Plotting Intercepts
Example continued:
y
(3, 6)
Plot all three of the
solutions (0, 0), (3, 6)
and (2, 4).
Draw the line
that contains the
three points.
(2, 4)
(0, 0)
Martin-Gay, Beginning and Intermediate Algebra, 4ed
x
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Vertical and Horizontal Lines
Vertical Lines
The graph of x = c, where c is a real number is
a vertical line with x-intercept (c, 0).
Horizontal Lines
The graph of y = c, where c is a real number, is
a horizontal line with y-intercept (0, c).
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Graph by Plotting Intercepts
Example:
Graph y = 3
Note that this line can be written as y = 0·x + 3
The y-intercept is (0, 3), but there is no x-intercept.
(Since an x-intercept would be found by letting y = 0,
and 0  0·x + 3, there is no x-intercept.)
Every value we substitute for x gives a y-coordinate
of 3.
The graph will be a horizontal line through the
point (0,3) on the y-axis.
Continued.
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Graph by Plotting Intercepts
y
Example continued:
y=3
(0, 3)
x
Martin-Gay, Beginning and Intermediate Algebra, 4ed
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Graph by Plotting Intercepts
Example:
Graph x = – 3.
This equation can be written x = 0·y – 3.
When y = 0, x = – 3, so the x-intercept is (– 3,0), but
there is no y-intercept.
Any value we substitute for y gives an x-coordinate
of –3.
So the graph will be a vertical line through the point
(– 3,0) on the x-axis.
Continued.
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Graph by Plotting Intercepts
y
Example continued:
x=–3
(– 3, 0)
Martin-Gay, Beginning and Intermediate Algebra, 4ed
x
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