Finding Intercepts of Rational Fractions
To find the y-intercept(s) (the point where the graph crosses the y-axis), substitute in 0 for x and solve for y or f(x).
To find the x-intercept(s) (the point where the graph crosses the x-axis â also known as zeros), substitute in 0 for y and solve for x.
Examples: Find the intercepts of the function given.
![](/math/calculus/images/finding_intercepts_of_rational_fractions_img1.png)
To find the y-intercept, we must substitute in 0 for each x:
![](/math/calculus/images/finding_intercepts_of_rational_fractions_img2.png)
And then simplify:
![](/math/calculus/images/finding_intercepts_of_rational_fractions_img3.png)
![](/math/calculus/images/finding_intercepts_of_rational_fractions_img4.png)
There is a y-intercept at
. (Notice that 0 is the x coordinate because on the y-axis, x = 0.)
To find the x-intercept, we must substitute in 0 for y or f(x):
![](/math/calculus/images/finding_intercepts_of_rational_fractions_img6.png)
And then solve by cross-multiplying:
![](/math/calculus/images/finding_intercepts_of_rational_fractions_img7.png)
0 = x + 10
x = -10
There is a y-intercept at
. (Notice that 0 is the y coordinate because on the x-axis, y = 0.)
![](/math/calculus/images/finding_intercepts_of_rational_fractions_img2.png)
And then simplify:
![](/math/calculus/images/finding_intercepts_of_rational_fractions_img3.png)
![](/math/calculus/images/finding_intercepts_of_rational_fractions_img4.png)
There is a y-intercept at
![](/math/calculus/images/finding_intercepts_of_rational_fractions_img5.png)
To find the x-intercept, we must substitute in 0 for y or f(x):
![](/math/calculus/images/finding_intercepts_of_rational_fractions_img6.png)
And then solve by cross-multiplying:
![](/math/calculus/images/finding_intercepts_of_rational_fractions_img7.png)
0 = x + 10
x = -10
There is a y-intercept at
![](/math/calculus/images/finding_intercepts_of_rational_fractions_img8.png)
![](/math/calculus/images/finding_intercepts_of_rational_fractions_img9.png)
To find the y-intercept, we must substitute in 0 for each x:
![](/math/calculus/images/finding_intercepts_of_rational_fractions_img10.png)
And then simplify:
![](/math/calculus/images/finding_intercepts_of_rational_fractions_img11.png)
There is a y-intercept at
.
To find the x-intercept, we must substitute in 0 for y or f(x):
![](/math/calculus/images/finding_intercepts_of_rational_fractions_img13.png)
And then solve by cross-multiplying:
![](/math/calculus/images/finding_intercepts_of_rational_fractions_img14.png)
![](/math/calculus/images/finding_intercepts_of_rational_fractions_img15.png)
We must now solve the quadratic either by factoring or by using the quadratic formula.
We can factor this trinomial, so we'll use that method:
![](/math/calculus/images/finding_intercepts_of_rational_fractions_img16.png)
![](/math/calculus/images/finding_intercepts_of_rational_fractions_img17.png)
![](/math/calculus/images/finding_intercepts_of_rational_fractions_img18.png)
There are y-intercepts at
.
![](/math/calculus/images/finding_intercepts_of_rational_fractions_img10.png)
And then simplify:
![](/math/calculus/images/finding_intercepts_of_rational_fractions_img11.png)
There is a y-intercept at
![](/math/calculus/images/finding_intercepts_of_rational_fractions_img12.png)
To find the x-intercept, we must substitute in 0 for y or f(x):
![](/math/calculus/images/finding_intercepts_of_rational_fractions_img13.png)
And then solve by cross-multiplying:
![](/math/calculus/images/finding_intercepts_of_rational_fractions_img14.png)
![](/math/calculus/images/finding_intercepts_of_rational_fractions_img15.png)
We must now solve the quadratic either by factoring or by using the quadratic formula.
We can factor this trinomial, so we'll use that method:
![](/math/calculus/images/finding_intercepts_of_rational_fractions_img16.png)
![](/math/calculus/images/finding_intercepts_of_rational_fractions_img17.png)
![](/math/calculus/images/finding_intercepts_of_rational_fractions_img18.png)
There are y-intercepts at
![](/math/calculus/images/finding_intercepts_of_rational_fractions_img19.png)
Note: Not all rational functions have both an x or y intercept. If you cannot find a real solution, then it does not have that intercept.
Practice: Find the x and y intercepts of each rational function:
![](/math/calculus/images/finding_intercepts_of_rational_fractions_img20.png)
![](/math/calculus/images/finding_intercepts_of_rational_fractions_img21.png)
![](/math/calculus/images/finding_intercepts_of_rational_fractions_img22.png)
![](/math/calculus/images/finding_intercepts_of_rational_fractions_img23.png)
![](/math/calculus/images/finding_intercepts_of_rational_fractions_img24.png)
Answers: 1)x-int.
![](/math/calculus/images/finding_intercepts_of_rational_fractions_img25.png)
![](/math/calculus/images/finding_intercepts_of_rational_fractions_img26.png)
![](/math/calculus/images/finding_intercepts_of_rational_fractions_img27.png)
![](/math/calculus/images/finding_intercepts_of_rational_fractions_img28.png)
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