Graphing Rational Functions and Identifying Asymptotes

Rational Functions

Graphing Rational Functions and Finding Asymptotes

Rational functions are functions of the form f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomials and Q(x) ≠ 0. Understanding their graphs involves identifying key features such as asymptotes and intercepts.

• Vertical Asymptotes: Occur at values of x where the denominator Q(x) = 0 and the numerator P(x) ≠ 0.

• Horizontal Asymptotes: Determined by the degrees of P(x) and Q(x). If degrees are equal, the asymptote is y = leading coefficient of P(x) divided by leading coefficient of Q(x).

• Intercepts: The y-intercept is found by evaluating f(0). The x-intercept(s) are found by solving P(x) = 0.

Example: Graphing $f(x) = \frac{x-2}{x+3}$

• Vertical Asymptote: Set denominator to zero: $x+3=0 \implies x=-3$

• Horizontal Asymptote: Degrees of numerator and denominator are both 1. Asymptote is $y=1$ (since leading coefficients are both 1).

• x-intercept: Set numerator to zero: $x-2=0 \implies x=2$

• y-intercept: $f(0) = \frac{0-2}{0+3} = -\frac{2}{3}$

To graph the function:

1. Draw the vertical asymptote at $x=-3$ (dashed line).

2. Draw the horizontal asymptote at $y=1$ (dashed line).

3. Plot the intercepts: $(2,0)$ and $(0,-\frac{2}{3})$.

4. Choose additional points on either side of the vertical asymptote to sketch the curve.

Summary Table:

Feature	Value
Vertical Asymptote	$x = -3$
Horizontal Asymptote	$y = 1$
x-intercept	$(2, 0)$
y-intercept	$(0, -\frac{2}{3})$

Additional info: When graphing rational functions, always check for holes (removable discontinuities) by factoring numerator and denominator and canceling common factors. In this example, there are no holes.