x and y-intercepts
Summary: Finding x & y intercepts
A rational function is a function of the form
R x = p x q x
R x
p x
q x
q ≠ 0 q
0
The domain is all real numbers except for numbers that make the denominator = 0.
x-intercepts are the points at which the graph crosses the x-axis. They are also known as roots, zeros, or solutions.
To find x-intercepts, let y (or f(x)) = 0 and solve for x. In rational functions, this means that you are multiplying by 0 so to find the x-intercept, just set the numerator (the top of the fraction) equal to 0 and solve for x.
Remember: x-intercepts are points that look like (x,0)
Example 1
For
y = x - 1 x - 2 y
x
1
x
2
The x-intercept is (1,0) since
x - 1 = 0 x
1
0
x = 1 x
1
The y-intercept is the point where the graph crosses the y-axis. If the graph is a function, there is only one y-intercept (and it only has ONE name)
To find the y-intercept (this is easier than the x-intercept), let x = 0. Plug in 0 for x in the equation and simplify.
Remember: y-intercepts are points that look like (0,y)
Example 2
For
y = x + 1 x - 2 y
x
1
x
2
The y-intercept is (0,
-1 2
-1
2
0 + 1 0 - 2 = -1 2
0 1 0 2
-1
2
Find the x- and y-intercepts of the following:
Problem 1
y = 1 x + 2 y
1
x
2
Solution 1
x-intercept: None since
1 ≠ 0 1
0
y-intercept: (0, 1 2 1 2
1 0 + 2 = 1 2 1
0
2
1
2
Problem 2
y = 1 - 3 x 1 - x y
1
3
x
1
x
Solution 2
x-intercept: (1 3 1 3
1 - 3 x = 0 1
3
x
0
-3 x = -1
-3
x
-1
x = 1 3 x
1
3
y-intercept: (0,1) since
1 - 3 × 0 1 - 0 = 1 1
3
0
1
0
1
Problem 3
y = x 2 x 2 + 9 y
x
2
x
2
9
Solution 3
x-intercept: (0,0) since
x 2 = 0 x
2
0
x = 0 x
0
y-intercept: (0,0) since
0 2 0 2 + 9 = 0 9 = 0 0
2
0
2
9
0
9
0
Problem 4
y = x + 1 x - 2 2 y
x
1
x
2
2
Solution 4
x-intercept: (-1,0) since
x + 1 = 0 x
1
0
x + 1 = 0 x
1
0
x = - 1 x
1
y-intercept: (0, 1 4 1 4
0 + 1 0 - 2 2 = 1 - 2 2 = 1 4 0
1
0
2
2
1
2
2
1
4
Problem 5
y = 3 x x 2 - x - 2 y
3
x
x
2
x
2
Solution 5
x-intercept: (0,0) since
3 x = 0 3
x
0
x = 0 x
0
y-intercept: (0,0) since the x-intercept is (0,0)
Problem 6
y = 1 x - 3 + 1 y
1
x
3
1
Solution 6
x-intercept: (2,0) since
1 x - 3 + 1 = 0 1
x
3
1
0
1 x - 3 = - 1 1
x
3
1
- x + 3 = 1 x
3
1
- x = -2 x
-2
x = 2 x
2
y-intercept: (0,
2 3 2
3
1 0 - 3 + 1 = -1 3 + 1 = 2 3 1
0
3
1
-1
3
1
2
3
Problem 7
y = x 2 - 4 x + 1 y
x
2
4
x
1
Solution 7
x-intercepts: (-2,0), (2,0) since
x 2 - 4 = 0 x
2
4
0
x 2 = 4 x
2
4
x = -2 x
-2
x = 2 x
2
y-intercept: (0, -4)
since
0 2 - 4 0 + 1 = = - 4 1 = -4 0
2
4
0
1
4
1
-4
Problem 8
y = 4 + 5 x 2 + 2 y
4
5
x
2
2
Solution 8
x-intercept: None since
4 + 5 x 2 + 2 = 0 4
5
x
2
2
0
5 x 2 + 2 = -4 5
x
2
2
-4
-4 x 2 - 8 = 5 -4
x
2
8
5
-4 x 2 = 13 -4
x
2
13
x 2 = 13 4 x
2
13
4
y-intercept: (0,
13 2 13
2
4 + 5 0 2 + 2 = 4 + 5 2 = 13 2 4
5
0
2
2
4
5
2
13
2
Problem 9
y = 5 x - 2 x - 3 y
5
x
2
x
3
Solution 9
x-intercept: (
2 5 2
5
5 x - 2 = 0 5
x
2
0
5 x - 2 = 0 5
x
2
0
5 x = 2 5
x
2
x = 2 5 x
2
5
y-intercept: None
since
y = 5 × 0 - 2 0 - 3 y
5
0
2
0
3
Problem 10
y = x 3 - 8 x 2 + 1 y
x
3
8
x
2
1
Solution 10
x-intercept: (2,0) since
x 3 - 8 = 0 x
3
8
0
x - 2 x 2 + 2 x + 4 = 0 x
2
x
2
2
x
4
0
x = 2 x
2
y-intercept: (0,-8)
since
0 3 - 8 0 2 + 1 = - 8 1 = -8 0
3
8
0
2
1
8
1
-8
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This work is licensed by Pradnya Bhawalkar and Kim Johnston. See the Creative Commons License about permission to reuse this material.
Last edited by Pradnya Bhawalkar on May 1, 2006 6:45 pm GMT-5.