7.2 Rational Functions and Their Graphs

Rational Functions and Their Graphs

A rational function is the quotient of two polynomials. These functions introduce behavior you don't see with regular polynomials: asymptotes, holes, and dramatic changes in output near certain x-values. Understanding how to find and interpret these features is central to graphing rational functions and analyzing their behavior.

more resources to help you study

practice questions

Rational Functions: Domain, Range, and Asymptotes

Definition and Characteristics of Rational Functions

A rational function has the form $f(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials and $Q(x) \neq 0$.

The domain is all real numbers except values that make the denominator zero. These are called non-permissible values. To find them, set $Q(x) = 0$ and solve.

What happens at a non-permissible value depends on whether the factor cancels with the numerator:

• If the factor does not cancel, you get a vertical asymptote (the function blows up toward $\pm\infty$).
• If the factor does cancel, you get a hole (a single missing point on an otherwise continuous curve).

The range is all real y-values the function actually outputs. Horizontal asymptotes and holes can create y-values the function never reaches, so you need to check for those.

Determining Horizontal and Oblique Asymptotes

Horizontal and oblique asymptotes describe end behavior, what the function does as $x \to \pm\infty$. The rule depends on comparing the degree of the numerator ($n$) to the degree of the denominator ($m$):

• If $n < m$: The horizontal asymptote is $y = 0$.
• If $n = m$: The horizontal asymptote is $y = \frac{a}{b}$, where $a$ and $b$ are the leading coefficients of the numerator and denominator.
• If $n = m + 1$: There's no horizontal asymptote. Instead, the function has an oblique (slant) asymptote, found by performing polynomial long division. The quotient (ignoring the remainder) gives you the equation of the slant asymptote.

Example: For $f(x) = \frac{2x^2 + 3x - 1}{x - 4}$, the numerator has degree 2 and the denominator has degree 1, so $n = m + 1$. Performing long division:

$2x^2 + 3x - 1 \div (x - 4) = 2x + 11 + \frac{43}{x - 4}$

The oblique asymptote is $y = 2x + 11$. (As $x \to \pm\infty$, the remainder fraction shrinks to zero, so the function hugs this line.)

Graphing Rational Functions: Key Features

Identifying Intercepts, Asymptotes, and Holes

Here's a step-by-step process for finding all the key features:

1. Factor the numerator and denominator completely.
2. Find holes: Identify any common factors that cancel. Set the cancelled factor equal to zero to get the x-coordinate of the hole. Plug that x-value into the simplified function to get the y-coordinate.
3. Find vertical asymptotes: After cancelling, set the remaining denominator equal to zero.
4. Find x-intercepts: Set the numerator of the simplified function equal to zero and solve.
5. Find the y-intercept: Evaluate $f(0)$ (plug in $x = 0$).
6. Find horizontal or oblique asymptotes using the degree comparison rules above.

Example: Consider $f(x) = \frac{x^2 - 4}{x - 2}$.

Factor the numerator: $\frac{(x-2)(x+2)}{x-2}$. The $(x-2)$ cancels, so the simplified form is $f(x) = x + 2$ with a hole at $x = 2$.

• Hole: At $x = 2$, the simplified function gives $y = 2 + 2 = 4$, so the hole is at $(2, 4)$.
• Vertical asymptote: None. The only non-permissible value produced a hole, not an asymptote.
• x-intercept: $x + 2 = 0 \Rightarrow x = -2$, so the x-intercept is $(-2, 0)$.
• y-intercept: $f(0) = 0 + 2 = 2$, so the y-intercept is $(0, 2)$.

The graph is just the line $y = x + 2$ with a single point removed at $(2, 4)$. There is no vertical asymptote here, and the function is neither even nor odd.

Sketching the Graph and Analyzing Behavior

Once you've plotted intercepts, asymptotes, and holes, fill in the curve:

1. Create a sign chart. The vertical asymptotes and x-intercepts divide the x-axis into intervals. Test a point in each interval to determine whether the function is positive or negative there.
2. Check behavior near vertical asymptotes. Plug in values just to the left and right of each asymptote to see whether the function heads toward $+\infty$ or $-\infty$ on each side.
3. Sketch the curve so it passes through your intercepts, approaches asymptotes correctly, and matches the signs from your chart.
4. Check for symmetry if it helps: if $f(-x) = f(x)$, the graph is symmetric about the y-axis (even). If $f(-x) = -f(x)$, it's symmetric about the origin (odd).

Analyzing Rational Functions: Limits and Asymptotes

Using Limits to Analyze Behavior Near Discontinuities

Limits formalize what's happening near non-permissible values.

For a hole at $x = a$: the limit $\lim_{x \to a} f(x)$ exists and equals the y-coordinate of the hole. The function just isn't defined there.

For a vertical asymptote at $x = a$: the function grows without bound. You check the one-sided limits:

• $\lim_{x \to a^-} f(x)$ (approaching from the left)
• $\lim_{x \to a^+} f(x)$ (approaching from the right)

Each one-sided limit will be $+\infty$ or $-\infty$. If they differ in sign, the function shoots in opposite directions on either side of the asymptote.

Example: For $f(x) = \frac{1}{x - 2}$:

• $\lim_{x \to 2^-} f(x) = -\infty$ (values just left of 2 make the denominator a small negative number)
• $\lim_{x \to 2^+} f(x) = +\infty$ (values just right of 2 make the denominator a small positive number)

This confirms a vertical asymptote at $x = 2$.

Determining Horizontal and Oblique Asymptotes Using Limits

You can also express the asymptote rules as limits:

• Horizontal asymptote: $\lim_{x \to \infty} f(x) = L$ means $y = L$ is a horizontal asymptote. Check $x \to -\infty$ as well (they're usually the same for rational functions, but verify).
• Oblique asymptote: If you found the slant asymptote $y = mx + b$ by long division, you can confirm it by showing $\lim_{x \to \infty} [f(x) - (mx + b)] = 0$.

Example: For $f(x) = \frac{2x^2 + 3x - 1}{x - 4}$, long division gave $y = 2x + 11$. Confirming:

$\lim_{x \to \infty} \left[\frac{2x^2 + 3x - 1}{x - 4} - (2x + 11)\right] = \lim_{x \to \infty} \frac{43}{x - 4} = 0$

So $y = 2x + 11$ is the oblique asymptote.

Transformations of Rational Functions

Types of Transformations and Their Representations

Transformations work the same way on rational functions as on any other function family:

• Horizontal shift: $f(x - h)$ shifts right $h$ units; $f(x + h)$ shifts left $h$ units
• Vertical shift: $f(x) + k$ shifts up $k$ units; $f(x) - k$ shifts down $k$ units
• Vertical stretch/compression: $af(x)$, where $|a| > 1$ stretches and $0 < |a| < 1$ compresses vertically
• Horizontal stretch/compression: $f(bx)$, where $|b| > 1$ compresses and $0 < |b| < 1$ stretches horizontally
• Reflections: $-f(x)$ reflects across the x-axis; $f(-x)$ reflects across the y-axis

Effects of Transformations on Key Features of the Graph

Every key feature moves with the transformation. Vertical asymptotes shift horizontally with horizontal transformations. Horizontal asymptotes shift vertically with vertical transformations. Holes move according to both.

Example: Start with the parent function $f(x) = \frac{1}{x}$, which has a vertical asymptote at $x = 0$ and a horizontal asymptote at $y = 0$.

The transformation $g(x) = \frac{-2}{x - 3} + 1$ can be read as $g(x) = -2f(x - 3) + 1$:

• Shift right 3: vertical asymptote moves to $x = 3$
• Vertical stretch by 2 and reflection across x-axis: the graph flips and stretches
• Shift up 1: horizontal asymptote moves to $y = 1$

The overall shape (two branches in opposite quadrants relative to the asymptotes) is preserved, but the location and orientation change.