Sign of the function, intercepts and asymptotes of the rational function
Before sketching any rational function it is worth asking three questions: where does it cross the axes? Where is it positive and where is it negative? Where does it "escape" to when $x$ approaches the excluded values or infinity? Answering these three questions —without needing a graphing calculator— is the aim of this section.
Intercepts with the axes
Calculate the axis intercepts of the function $f(x) = \frac{2x-6}{x+4}$.
Calculate the axis intercepts of the function $f(x) = \frac{2x-6}{x+4}$.
Sign of the rational function
The sign of $\dfrac{P(x)}{Q(x)}$ can only change at the points where $P$ or $Q$ is zero (the "critical points"). Between two consecutive critical points, the sign is constant.
Determine the intervals in which the function $f(x) = \frac{x-3}{x+2}$ takes strictly positive values.
Determine the intervals in which the function $f(x) = \frac{x-3}{x+2}$ takes strictly positive values.
Analytically study the sign of the function $f(x) = \frac{-2x}{x^2-4}$ over its entire domain.
Analytically study the sign of the function $f(x) = \frac{-2x}{x^2-4}$ over its entire domain.
Vertical asymptotes
Find the domain of definition and the equations of the vertical asymptotes of the function $f(x) = \frac{x+7}{x^2-9}$.
Find the domain of definition and the equations of the vertical asymptotes of the function $f(x) = \frac{x+7}{x^2-9}$.
Horizontal or oblique asymptote according to the degrees
The behaviour at infinity of $\dfrac{P(x)}{Q(x)}$ depends on comparing the degree of $P$ with that of $Q$, just as with limits of rational sequences.
Find the equations of the vertical and horizontal asymptotes of the rational function $f(x) = \frac{5x+1}{2x-4}$.
Find the equations of the vertical and horizontal asymptotes of the rational function $f(x) = \frac{5x+1}{2x-4}$.
Calculate the equation of the oblique asymptote of the function $f(x) = \frac{x^2-5x+6}{x-1}$.
Calculate the equation of the oblique asymptote of the function $f(x) = \frac{x^2-5x+6}{x-1}$.
Given the function $f(x) = \frac{3x^2-2}{x^2+1}$, determine the equation of its horizontal asymptote by evaluating the relevant limits as $x \to +\infty$ and…
Given the function $f(x) = \frac{3x^2-2}{x^2+1}$, determine the equation of its horizontal asymptote by evaluating the relevant limits as $x \to +\infty$ and $x \to -\infty$.
Reading a rational function completely
Putting it all together: intercepts, sign and asymptotes allow us to describe the shape of the graph without having to tabulate many points.
Determine the numerical values of the parameters $a$ and $b$ knowing that the function $f(x) = \frac{ax+3}{bx-2}$ has a vertical asymptote at the line $x = 1$ and a…
Determine the numerical values of the parameters $a$ and $b$ knowing that the function $f(x) = \frac{ax+3}{bx-2}$ has a vertical asymptote at the line $x = 1$ and a horizontal asymptote at the line $y = 4$.
Find the asymptotes and axis intercepts of the function $f(x) = \frac{x^3-8}{x^2-4}$. Pay special attention to possible removable discontinuities when simplifying the…
Find the asymptotes and axis intercepts of the function $f(x) = \frac{x^3-8}{x^2-4}$. Pay special attention to possible removable discontinuities when simplifying the expression.
Discuss, depending on the values of the real parameter $k$, the number and type of vertical asymptotes of the function $f(x) = \frac{x^2-x-2}{x^2-k}$. Also determine the…
Discuss, depending on the values of the real parameter $k$, the number and type of vertical asymptotes of the function $f(x) = \frac{x^2-x-2}{x^2-k}$. Also determine the exact value of $k$ for which the function has a removable discontinuity.