Sign, Intercepts, and Asymptotes of a Rational Function
Before graphing a rational function, it helps to ask yourself three questions: Where does it cross the axes? Where is it positive and where is it negative? Where does it "escape" to when $x$ gets close to a forbidden value or to infinity? With the answers you can draw the graph without making a table of dozens of points. That is what we will do on this page.
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 a rational function
The sign of $\dfrac{P(x)}{Q(x)}$ can only change at the points where $P$ or $Q$ equals 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.
The efficiency of a modified thermal cycle, as a function of the compression ratio $v > 1$, is given by the model $E(v) = \frac{v^2-3v+2}{v^2+v}$. Analyse the sign of…
The efficiency of a modified thermal cycle, as a function of the compression ratio $v > 1$, is given by the model $E(v) = \frac{v^2-3v+2}{v^2+v}$. Analyse the sign of the efficiency to determine the range of $v$ in which the cycle produces useful work (that is, when $E(v) > 0$) and find the theoretical limiting efficiency when $v \to +\infty$.
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 slant asymptote
The behavior of $\dfrac{P(x)}{Q(x)}$ at infinity depends on comparing the degree of $P$ with that of $Q$.
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 $x \to -\infty$.
Position of the curve relative to the slant asymptote
A curve approaches its slant asymptote, but does it do so from above or from below? That is decided by the sign of the difference between the function and the line.
Calculate the equation of the oblique asymptote of the function $f(x) = \frac{2x^2+1}{x-1}$ and analytically study the relative position of the curve with respect to…
Calculate the equation of the oblique asymptote of the function $f(x) = \frac{2x^2+1}{x-1}$ and analytically study the relative position of the curve with respect to this asymptote as $x \to +\infty$ and as $x \to -\infty$.
Fully investigate the asymptotes and the relative position of the curve with respect to them for the function $f(x) = \frac{2x^3-5x^2+4x+1}{x^2-2x+1}$. Take into account…
Fully investigate the asymptotes and the relative position of the curve with respect to them for the function $f(x) = \frac{2x^3-5x^2+4x+1}{x^2-2x+1}$. Take into account any potential multiple roots that may appear in the denominator when calculating the limits.
Complete analysis of a rational function
By putting together intercepts, sign, and asymptotes, you can describe the shape of the graph. First, always factor and simplify.
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.
Parameters and asymptotes
Sometimes you are given the asymptotes or the intercepts and asked for the coefficients. The idea is always the same: translate each piece of data into an equation about the parameters and solve them.
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 values of the parameters $a$, $b$ and $c$ such that the rational function $f(x) = \frac{ax^2+bx+c}{x+2}$ has an oblique asymptote with equation $y = 3x - 1$ and…
Find the values of the parameters $a$, $b$ and $c$ such that the rational function $f(x) = \frac{ax^2+bx+c}{x+2}$ has an oblique asymptote with equation $y = 3x - 1$ and crosses the x-axis at exactly the point $x = 1$.
Given the rational function $f(x) = \frac{x^3+kx^2+x-1}{x^2-1}$, find the value of the parameter $k$ such that the curve representing the function crosses its own…
Given the rational function $f(x) = \frac{x^3+kx^2+x-1}{x^2-1}$, find the value of the parameter $k$ such that the curve representing the function crosses its own oblique asymptote at the point with x-coordinate $x = 2$.
Discussion according to a parameter
When a parameter $k$ appears in the denominator, the number of vertical asymptotes depends on $k$. We study the cases.
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.
Reading a graph
You can also go in the opposite direction: from the graph to the formula. You read the asymptotes and the intercepts, and with them you set up equations.
Observe the following graph of a rational function and, from it, determine the equations of its asymptotes and the coordinates of its axis intercepts.
Observe the following graph of a rational function and, from it, determine the equations of its asymptotes and the coordinates of its axis intercepts.
From the graph of the rational function $f(x)$ shown below, deduce its algebraic expression knowing that it is of the form $f(x) = \frac{ax+b}{cx+d}$.
From the graph of the rational function $f(x)$ shown below, deduce its algebraic expression knowing that it is of the form $f(x) = \frac{ax+b}{cx+d}$.
The following graph shows a rational function that has an oblique asymptote and a vertical asymptote. Given that the function is of the form…
The following graph shows a rational function that has an oblique asymptote and a vertical asymptote. Given that the function is of the form $f(x) = \frac{x^2+ax+b}{x+c}$, analyse its visual features (asymptotes and intercepts) to determine the value of the coefficients $a$, $b$ and $c$.
Applications: long-term trends
Many real-world models are rational functions: populations, drug concentrations, average costs. Asymptotes tell you what happens "in the long run."
The population of an algae species in a pond, measured in thousands of individuals, evolves according to the function $P(t) = \frac{120t+50}{2t+5}$, where $t \geq 0$ is…
The population of an algae species in a pond, measured in thousands of individuals, evolves according to the function $P(t) = \frac{120t+50}{2t+5}$, where $t \geq 0$ is the time elapsed in days. Calculate the initial population, the moment at which the population reaches $40$ thousand individuals, and the long-term limit of the population (horizontal asymptote).
The total cost of producing $x$ hundred pieces of an electronic component is $C(x) = 60x^2+100x+240$ hundred pesos. If the average cost function per hundred pieces is…
The total cost of producing $x$ hundred pieces of an electronic component is $C(x) = 60x^2+100x+240$ hundred pesos. If the average cost function per hundred pieces is defined as $C_m(x) = \frac{C(x)}{x}$, find the oblique asymptote of $C_m(x)$ and explain its economic meaning for very large levels of production.
A demographic model establishes that the growth rate of a population, measured as an annual percentage, is given by the function $T(x) = \frac{x^2-8x+15}{x^2+4}$, where…
A demographic model establishes that the growth rate of a population, measured as an annual percentage, is given by the function $T(x) = \frac{x^2-8x+15}{x^2+4}$, where $x \geq 0$ is the time elapsed in decades. Study the sign of $T(x)$ to determine the historical periods of population growth and decline, and calculate the horizontal asymptote to interpret the limiting long-term trend of the rate.
Applications: sign, intercepts, and asymptotes in a model
In a model with $t\ge0$, the sign tells you when the quantity is positive (makes physical sense) and the horizontal asymptote tells you what happens in the end.
Back to the beginning: with $f(0)$ and $P(x)=0$ you find the intercepts; with the critical points, the sign; and with $Q(x)=0$ and the degrees, the asymptotes.
The concentration of a pollutant gas in a test chamber is given by the expression $C(t) = \frac{5t}{t^2+4}$ parts per million (ppm), with $t \geq 0$ measured in hours.…
The concentration of a pollutant gas in a test chamber is given by the expression $C(t) = \frac{5t}{t^2+4}$ parts per million (ppm), with $t \geq 0$ measured in hours. Determine the moments at which the concentration is zero, study the sign of $C(t)$ to interpret its physical meaning, and find the horizontal asymptote stating what happens in the long term.
The amount of a drug present in the bloodstream, in milligrams, $t$ hours after oral administration is modelled by the function $M(t) = \frac{50t^2}{t^3+8}$. Study the…
The amount of a drug present in the bloodstream, in milligrams, $t$ hours after oral administration is modelled by the function $M(t) = \frac{50t^2}{t^3+8}$. Study the axis intercepts, the sign of the function for $t \geq 0$, and its asymptotic behavior.
The deflection of a supported beam under a variable load can be approximated by the function $D(x) = \frac{x^3-4x^2+3x}{x^2+1}$, where $x$ is the distance in meters…
The deflection of a supported beam under a variable load can be approximated by the function $D(x) = \frac{x^3-4x^2+3x}{x^2+1}$, where $x$ is the distance in meters measured from the left end of the support. Determine the points where the deflection is zero, study the sign of $D(x)$ in the interval $x \in [0, 4]$ and calculate the equation of the oblique asymptote of the function.