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Rational Functions
& Asymptotes

Learn to graph functions of the form \(\frac{p(x)}{q(x)}\), identify vertical, horizontal, and oblique asymptotes, and find zeros from factored forms.

Algebra 2

What Is a Rational Function?

A rational function is any function that can be written as the ratio of two polynomials:

The domain excludes every \(x\)-value that makes the denominator zero. These excluded values are where the most interesting behavior happens: vertical asymptotes and holes.

The Parent Function \(f(x)=\tfrac{1}{x}\)

\(x\) \(y\) \(1\) \(-1\) \(1\) \(-1\) \(y = \tfrac{1}{x}\) VA: \(x = 0\) HA: \(y = 0\)

Three Types of Asymptotes

Vertical
Occurs at values where the denominator equals zero (and the factor does not cancel). The graph shoots toward \(\pm\infty\).
Horizontal
Describes end behavior as \(x \to \pm\infty\). Determined by comparing the degrees of the numerator and denominator.
Oblique (Slant)
Occurs when the numerator's degree is exactly one more than the denominator's. Found by polynomial long division.

Finding Vertical Asymptotes

1
Factor numerator and denominator completely.
2
Cancel any common factors: those give holes, not VAs.
3
Solve the remaining denominator = 0. Each solution \(x = a\) is a vertical asymptote.

Horizontal Asymptote Rules

Let \(n\) = degree of the numerator and \(m\) = degree of the denominator.

Condition Horizontal Asymptote End Behavior
\(n < m\) \(y = 0\) Graph flattens to the \(x\)-axis
\(n = m\) \(y = \frac{a_n}{b_m}\) Ratio of leading coefficients
\(n > m\) No HA Oblique or polynomial end behavior

VA vs HA vs Hole

Feature Vertical Asymptote Horizontal Asymptote Hole
Direction Vertical line \(x = a\) Horizontal line \(y = b\) Single missing point
Cause Denominator = 0 (non-cancelable) Degree comparison Common factor cancels
Graph crosses it? Never Sometimes N/A (point removed)
How to find Solve \(q(x)=0\) Compare degrees Cancel common factors

Finding the Zeros

1
Factor the numerator \(p(x)\).
2
Set each factor equal to zero and solve.
3
Check that none of these solutions also make the denominator zero. If one does, that's a hole, not a zero.

How to Graph a Rational Function

1
Factor numerator and denominator completely.
2
Identify holes: cancel common factors, note the \(x\)-value.
3
Find vertical asymptotes: set remaining denominator = 0.
4
Find horizontal/oblique asymptote: compare degrees.
5
Find zeros: set simplified numerator = 0.
6
Find y-intercept: evaluate \(f(0)\).
7
Plot a few test points in each region, then sketch the curve.

Example 1: \(f(x) = \dfrac{2x}{x-1}\)

\(1\) \(2\) \(3\) \(4\) \(-1\) \(-2\) \(1\) \(2\) \(-1\) \(x = 1\) \(y = 2\) \((0,0)\)

Example 2: \(g(x) = \dfrac{x^2 - 4}{x^2 - x - 2}\)

VA
\(x = -1\) (remaining denominator = 0)
HA
\(y = 1\) (same degree, ratio \(\frac{1}{1}\))
Hole
At \(x = 2\): \(y = \frac{4}{3}\)
Zero
\(x = -2\) (numerator = 0)

Example 3: A Potential Oblique Asymptote

Graph \(h(x) = \dfrac{x^2 + 2x - 3}{x - 1}\). Since the numerator's degree (2) exceeds the denominator's degree (1) by exactly one, there is a potential for an oblique asymptote.

True Oblique: \(r(x) = \dfrac{x^2 + 1}{x}\)

\(x\) \(y\) \(1\) \(2\) \(3\) \(-1\) \(-2\) \(-3\) \(1\) \(2\) VA: \(x = 0\) OA: \(y = x\)

Understanding End Behavior

End behavior describes what happens to \(f(x)\) as \(x \to +\infty\) and \(x \to -\infty\). For rational functions, the asymptote tells the whole story:

HA exists
The graph approaches the horizontal line. Example: \(\frac{3x}{x+1} \to 3\) as \(x \to \pm\infty\).
OA exists
The graph approaches the slant line. The gap \(\frac{r}{q(x)} \to 0\) as \(x \to \pm\infty\).

Common Mistakes

✕ Forgetting to Factor First
Calling \(x = 2\) a VA in \(\frac{x-2}{(x-2)(x+1)}\) when it's actually a hole. Always factor and cancel before identifying asymptotes.
✕ Wrong HA Rule
Using the HA formula when the numerator's degree is larger. If \(n > m\), there is no horizontal asymptote: look for an oblique one instead.
✕ Zeros from the Denominator
Setting the denominator equal to zero to find zeros. Zeros come from the numerator. The denominator gives you asymptotes or holes.

Summary Cheat Sheet

Find... Method Quick Check
Zeros Set numerator = 0 (after canceling) Graph crosses \(x\)-axis
VA Set denominator = 0 (after canceling) Graph shoots to \(\pm\infty\)
Hole Canceled common factor: \(x = a\) Open circle on graph
HA Compare degrees \(n\) vs \(m\) Graph flattens at large \(|x|\)
OA Long division when \(n = m + 1\) Graph follows a slant line
y-intercept Evaluate \(f(0)\) Graph crosses \(y\)-axis

Question 1 of 5

What is the vertical asymptote of \(f(x) = \dfrac{3}{x + 5}\)?

Question 2 of 5

What is the horizontal asymptote of \(g(x) = \dfrac{4x^2 - 1}{2x^2 + 3}\)?

Question 3 of 5

For \(h(x) = \dfrac{(x-3)(x+2)}{(x-3)(x-1)}\), what occurs at \(x = 3\)?

Question 4 of 5

Which rational function has a slant (oblique) asymptote?

Question 5 of 5

Find all zeros of \(k(x) = \dfrac{x^2 - 9}{x^2 - 4x + 3}\).