9231 · 1.2
Rational functions and graphs
Rational functions are essentially fractions made of polynomials. Their graphs often have 'breaks' called asymptotes, which are invisible boundary lines that the curve approaches but (usually) never touches.
Need to know
What you need to know
- **Vertical Asymptotes (VA):** These occur where the denominator is zero (and the numerator is non-zero). If $y = \frac{P(x)}{Q(x)}$, the lines $x=a$ are vertical asymptotes for each root $a$ of $Q(x)=0$. The function is undefined at these x-values.
- **Horizontal Asymptotes (HA):** These describe the behaviour as $x \to \pm \infty$. Compare the degrees of the numerator (deg(P)) and denominator (deg(Q)).
- If deg(P) < deg(Q), the horizontal asymptote is the x-axis, $y=0$.
- If deg(P) = deg(Q), the horizontal asymptote is $y = \frac{\text{leading coefficient of P}}{\text{leading coefficient of Q}}$.
- If deg(P) > deg(Q), there is no horizontal asymptote. Look for an oblique one instead.
Explanation
Decoding Rational Graphs
- Find Asymptotes: Locate vertical asymptotes by setting the denominator to zero. Determine horizontal or oblique asymptotes by comparing the degrees of the numerator and denominator.
- Find Intercepts: Calculate the y-intercept by setting x=0. Find the x-intercept(s) by setting the numerator to zero.
- Find Stationary Points: Differentiate the function using the quotient rule, set the derivative to zero, and solve for x to find the coordinates of any turning points.
- Sketch and Refine: Draw the asymptotes as dashed lines and plot the intercepts and stationary points. Sketch the curve, checking its behaviour near the asymptotes to complete the graph.