9231 · 1.2
Rational functions and graphs flashcards
Revision flashcards for Cambridge 9231 Rational functions and graphs (syllabus 1.2). Flip, recall, then mark a real past-paper question.
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What is a rational function?
A function that can be expressed as a ratio of two polynomials, $f(x) = \frac{P(x)}{Q(x)}$, where $Q(x) \neq 0$.
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How do you find the equation of a vertical asymptote?
Set the denominator, $Q(x)$, equal to zero and solve for $x$. If $x=a$ is a solution and $P(a) \neq 0$, then $x=a$ is a vertical asymptote.
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When does a rational function have a horizontal asymptote at $y=0$?
When the degree of the numerator polynomial is less than the degree of the denominator polynomial. deg(P) < deg(Q).
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How do you find the horizontal asymptote when the degrees of the numerator and denominator are equal?
The horizontal asymptote is $y = \frac{a}{b}$, where $a$ is the leading coefficient of the numerator and $b$ is the leading coefficient of the denominator.
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What is an oblique (or slant) asymptote?
A non-vertical, non-horizontal line that the graph of a function approaches. It occurs in rational functions when the degree of the numerator is exactly one greater than the degree of the denominator.
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How do you find the equation of an oblique asymptote?
Use polynomial long division to divide the numerator by the denominator. The quotient (ignoring the remainder) gives the equation of the oblique asymptote, $y = ax+b$.
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Can the graph of a rational function ever cross a vertical asymptote?
No. A vertical asymptote corresponds to a value of $x$ for which the function is undefined.
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Can the graph of a rational function ever cross a horizontal or oblique asymptote?
Yes. This can happen for finite values of $x$. Asymptotes describe the end behaviour of the function as $x \to \pm\infty$.
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How do you find the point where a curve crosses its horizontal asymptote, $y=k$?
Set the function equal to the value of the asymptote and solve for $x$. That is, solve $\frac{P(x)}{Q(x)} = k$.
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Common Trap: What happens if a factor cancels from the numerator and denominator, e.g., $y = \frac{(x-1)(x+2)}{x-1}$?
The graph is $y=x+2$ but with a 'hole' or point of discontinuity at $x=1$, not a vertical asymptote. Always simplify the function first, but note any such excluded points.