Functions Reference

Rational Functions and Asymptotes

A rational function is a fraction of polynomials. The denominator controls restrictions and vertical behavior, while the degrees of the numerator and denominator guide end behavior.

Fact Table

Question	Where to Look	Result
What values are not in the domain?	Original denominator.	Set the denominator not equal to zero.
Where are the vertical asymptotes?	Denominator after canceling common factors.	Uncanceled denominator zeros give vertical asymptotes.
Where are the holes?	Factors that cancel.	The canceled x-value is a hole, not an asymptote.
What is the horizontal asymptote?	Degrees of numerator and denominator.	Compare degrees before long division.
Where are the x-intercepts?	Simplified numerator.	Zeros of the numerator, unless the factor canceled.
Where is the y-intercept?	Evaluate $f(0)$.	Use only if $0$ is in the domain.

Content Formulas

Rational Function

$$f(x)=\frac{p(x)}{q(x)},\quad q(x)\ne0$$

Domain Restrictions

$$q(x)\ne0$$

Horizontal Asymptotes

$$\deg p<\deg q:\ y=0$$ $$\deg p=\deg q:\ y=\frac{\text{lead }p}{\text{lead }q}$$

Slant Asymptote

$$\deg p=\deg q+1$$ $$\text{divide }p(x)\text{ by }q(x)$$

Factor first. Cancel only after recording the original restrictions, because canceled factors become holes.

Classic Examples

Vertical Asymptote and Hole

Find the domain restrictions, hole, and vertical asymptote for $f(x)=\frac{x^2-1}{x^2-3x+2}$.

Solution StrategyFactor numerator and denominator, record every denominator zero, then cancel common factors.

$$f(x)=\frac{(x-1)(x+1)}{(x-1)(x-2)}$$ $$x\ne1,\quad x\ne2$$ $$f(x)=\frac{x+1}{x-2},\quad x\ne1,\ x\ne2$$

The factor $x-1$ canceled, so $x=1$ is a hole. The factor $x-2$ remains in the denominator, so $x=2$ is a vertical asymptote.

$$y=\frac{1+1}{1-2}=-2$$ $$\text{hole at }(1,-2)$$ $$\text{vertical asymptote: }x=2$$

Horizontal Asymptote

Find the horizontal asymptote of $g(x)=\frac{3x^2-5x+1}{2x^2+7}$.

Solution StrategyCompare degrees. Equal degrees use the ratio of leading coefficients.

$$\deg(3x^2-5x+1)=2$$ $$\deg(2x^2+7)=2$$ $$y=\frac{3}{2}$$

Slant Asymptote

Find the slant asymptote of $h(x)=\frac{x^2+3x+1}{x+1}$.

Solution StrategyThe numerator degree is exactly one more than the denominator degree, so divide.

$$\frac{x^2+3x+1}{x+1}=x+2-\frac{1}{x+1}$$ $$\text{slant asymptote: }y=x+2$$

Intercepts

Find the intercepts of $r(x)=\frac{x-4}{x+2}$.

Solution StrategySet the numerator equal to zero for x-intercepts, and evaluate $r(0)$ for the y-intercept.

$$x-4=0$$ $$x=4$$ $$r(0)=\frac{0-4}{0+2}=-2$$ $$\text{x-intercept: }(4,0)$$ $$\text{y-intercept: }(0,-2)$$

Graphing Checklist

Step	Action
1	Factor numerator and denominator.
2	Record domain restrictions from the original denominator.
3	Cancel common factors and mark holes.
4	Use remaining denominator zeros for vertical asymptotes.
5	Use degree comparison or division for end behavior.
6	Find intercepts and use a sign chart to place branches.