M151Q Precalculus · Unit 3: Quadratics, polynomials, triangles · on Exam 3
MFG 7.4–7.5 Rational functions and equations with algebraic fractions
Study notes for Rational functions and equations with algebraic fractions in M151Q Precalculus at Montana State University: the big ideas, key formulas, a worked example, common mistakes and an exam tip. The same topic has endless practice questions and flashcards in Mathub.
Big ideas
- $f(x) = \dfrac{p(x)}{q(x)}$. Vertical asymptotes where $q = 0$ and $p \ne 0$ (after cancelling common factors; a cancelled factor makes a hole instead). $x$-intercepts where $p = 0$.
- Horizontal asymptote from the degrees: $\deg p < \deg q$: $y = 0$; equal: $y =$ ratio of leading coefficients; $\deg p > \deg q$: none (the graph follows the quotient).
- Sketch using intercepts, asymptotes, and a sign check in each interval. Near a vertical asymptote the graph shoots up or down; test a point on each side.
- Solve equations with fractions by multiplying through by the LCD, then check that no solution makes a denominator zero (extraneous solutions).
Key formulas
Asymptotes
$$\text{VA: } q(x) = 0;\qquad \text{HA: } \deg p<\deg q \Rightarrow y = 0;\ \deg p = \deg q \Rightarrow y = \frac{a_n}{b_n}$$
Clearing fractions
$$\frac{A}{x} + \frac{B}{x-2} = C \ \xrightarrow{\ \times x(x-2)\ }\ A(x-2) + Bx = Cx(x-2)$$
Worked example
Find the intercepts and asymptotes of $f(x) = \dfrac{2x^2 - 8}{x^2 - x - 6}$.
Show the solution
Factor: $\dfrac{2(x-2)(x+2)}{(x-3)(x+2)} = \dfrac{2(x-2)}{x-3}$ for $x \ne -2$: hole at $x=-2$, VA $x = 3$, $x$-intercept $2$, $y$-intercept $\tfrac{-8}{-6} = \tfrac43$, HA $y = 2$ (equal degrees, $2/1$).
Common mistakes
- Calling a hole a vertical asymptote.
- Reading the horizontal asymptote from the constant terms instead of the leading coefficients.
- Keeping an extraneous solution after clearing fractions.
Exam tipFactor everything first. Cancelled factors are holes; leftover denominator factors are asymptotes.
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