Rational Functions and Their Graphs

Rational function with hole (or point of discontinuity) at x=5

(x+2)(x-5)/(x-5); Because x=5 is a zero of both the numerator and the denominator, the graph has a hole at x=5.

Key Words

-Continuous Graph -Discontinuous Graph -Non-Removable Discontinuity -Point of Discontinuity -Rational Function -Removable Discontinuity

Objectives

-Identify properties of rational functions -Graph rational functions

Rational function with horizontal asymptote at y=3

6x+1/2x; The degree of the numerator and denominator are the same, so the horizontal asymptote is the coefficient of 6x divided by the coefficient of 2x, or 6/2. So the horizontal asymptote is y=3.

Hole in a Graph

A hole in a graph occurs when the point of discontinuity results in both the numerator and denominator being equal to zero. The x-coordinate for the hole will be at the factor that can be cancelled, in this case x=3. y=x^2-3x/x^2-9 y=x(x-3)/(x+3)(x-3)=x/x+3 To find the value of y, substitute 3 into the simplified function for x and solve. x=3, y=x/x+3=3/3+3=3/6=1/2 When the function is graphed, the hole is indicated by using a hollow dot. Hole at (3, 1/2)

Rational Function

A rational function is expressed as P of x over Q of x where P of x and Q of x are polynomial functions and Q of x is not equal to zero. f(x)=P(x)/Q(x), Q(x) is not equal to 0.

Point of Discontinuity

Any value of x that results in the denominator being equal to zero is called a point of discontinuity.

Graph the function y=x+3/x^2-6x+5

First, calculate the vertical and horizontal asymptote. x^2-6x+5=0 (x-1)(x-5)=0 x-1=0 and x-5=0 x-1=0 and x-5=0 x=1 and x=5 Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y=0. Next, calculate the intercepts. Find y-intercept: y=0+3/0^2-6(0)+5 y=3/5 The y-intercept is at (0, 3/5). Find x-intercept: 0=x+3/x^2-6x+5 0=x+3 -3=x Finally, create a table to plot points near the aymptote.

Graph the rational function y=2x+4/x-6

First, calculate the vertical and horizontal asymptote. To find the vertical asymptote, set the denominator equal to zero and solve for x. x-6=0 x=6 The vertical asymptote will be at x=6. To find the horizontal asymptote first, look at the degrees of the numerator and the denominator. If they have the same degree, the horizontal asymptote is found by dividing the coefficients of the leading terms. In this function, both the numerator and denominator are linear, so you will divide the leading coefficients. y=2/1 y=2 The horizontal asymptote will be at y=2.

X- and Y- Intercepts

Next, calculate the x- and y-intercepts. To find the y-intercept, substitute zero for x in the function and solve for y. To find the x-intercept, substitute zero for y and solve for x. Find y-intercept: y=2x+4/x-6 y=2(0)+4/0-6 y=-4/6 y=-2/3 Find x-intercept 0=2x+4/x-6 0=2x+4 -4=2x -2=x Finally, to determine the shape of the graph, create a table to plot points near the asumptotes. Be sure to choose values for x that are near the asymptote.

Graphing Rational Functions

When graphing rational functions, you find the asymptotes and intercepts and plot a few key points.

Rational function with vertical asymptote at x=-4

x/x+4; The zero of the denominator is -4. The zero of the numerator is not -4, so the zero indicates the vertical asymptote is at x=-4.