How to Find Intercepts in a Rational Function

A graph's intercepts tell you where it crosses the axes.
••• Jupiterimages/ Images

The intercepts of a function are the values of x when f(x) = 0 and the value of f(x) when x = 0, corresponding to the coordinate values of x and y where the graph of the function crosses the x- and y-axes. Find the y-intercept of a rational function as you would for any other type of function: plug in x = 0 and solve. Find the x-intercepts by factoring the numerator. Remember to exclude holes and vertical asymptotes when finding the intercepts.

    Plug the value x = 0 into the rational function and determine the value of f(x) to find the y-intercept of the function. For example, plug x = 0 into the rational function f(x) = (x^2 - 3x + 2) / (x - 1) to get the value (0 - 0 + 2) / (0 - 1), which is equal to 2 / -1 or -2 (if the denominator is 0, there is a vertical asymptote or hole at x = 0 and therefore no y-intercept). The y-intercept of the function is y = -2.

    Factor the numerator of the rational function completely. In the above example, factor the expression (x^2 - 3x + 2) into (x - 2)(x - 1).

    Set the factors of the numerator equal to 0 and solve for the value of the variable to find the potential x-intercepts of the rational function. In the example, set the factors (x - 2) and (x - 1) equal to 0 to get the values x = 2 and x = 1.

    Plug the values of x you found in Step 3 into the rational function to verify that they are x-intercepts. X-intercepts are values of x that make the function equal to 0. Plug x = 2 into the example function to get (2^2 - 6 + 2) / (2 - 1), which equals 0 / -1 or 0, so x = 2 is an x-intercept. Plug x = 1 into the function to get (1^2 - 3 + 2) / (1 - 1) to get 0 / 0, which means there is a hole at x = 1, so there is only one x-intercept, x = 2.

Related Articles

How to Know the Difference between a Vertical Asymptote,...
How to Find the X Intercept of a Function
How to Find the Zeros of a Function
How to Find Vertical & Horizontal Asymptotes
How to Find Asymptotes & Holes
How to Find Horizontal Asymptotes of a Graph of a Rational...
How to Find the Coordinates of a Hole in a Graph
How to Calculate Function From Ordered Pairs
How to Solve a Parabola
How to Calculate a Horizontal Tangent Line
How to Find Equations of Tangent Lines
How to Factor Cubic Trinomials
How to Find the Point of Discontinuity in Algebra II
How to Find Rational Zeros of Polynomials
How to Integrate Sin^2 X
How to Solve Trinomials With Fractional Exponents
How to Factor Polynomials With 4 Terms
How to Find Linear Functions
How to Factorise a Quadratic Expression