Rational Functions with Quadratics (DP IB Analysis & Approaches (AA): HL): Revision Note

Quadratic rational functions & graphs

How do I sketch the graph of a rational function where the terms are not linear?

• A rational function can be written $f\left(x\right)=\frac{g\left(x\right)}{h\left(x\right)}$

  • Where g and h are polynomials

• To find the y-intercept evaluate $\frac{g\left(0\right)}{h\left(0\right)}$

• To find the x-intercept(s) solve $g\left(x\right)=0$

• To find the equations of the vertical asymptote(s) solve $h\left(x\right)=0$

• There will also be an asymptote determined by what f(x) tends to as x approaches infinity

  • In this course it will be either:

    • Horizontal

    • Oblique (a slanted line)

  • This can be found by writing $g\left(x\right)$ in the form $h\left(x\right)Q\left(x\right)+r\left(x\right)$

    • You can do this by polynomial division or comparing coefficients

  • The function then tends to the curve $y=Q\left(x\right)$

What are the key features of rational graphs?

Quadratic over linear

• For the rational function of the form $f\left(x\right)=\frac{a{x}^2+bx+c}{dx+e}$

  • e.g. $f\left(x\right)=\frac{4x^2+7x-2}{2x+5}$

• The graph has a y-intercept at $(0,\frac{c}{e})$ provided $e\ne 0$

  • e.g. the y-intercept of $f\left(x\right)=\frac{4x^2+7x-2}{2x+5}$ is $\left(0,-\frac{2}{5}\right)$

  • e.g. $f\left(x\right)=\frac{4x^2+7x-2}{2x}$ does not have a y-intercept

• The graph can have 0, 1 or 2 roots

  • They are the solutions to $a{x}^2+bx+c=0$

    • e.g. $f\left(x\right)=\frac{4x^2+7x-2}{2x+5}$ has two roots $\left(\frac{1}{4},0\right)$ and $\left(-2,0\right)$

    • e.g. $f\left(x\right)=\frac{4x^2+4x+1}{2x+5}$ has one root $\left(-\frac{1}{2},0\right)$

    • e.g. $f\left(x\right)=\frac{4x^2+1}{2x+5}$ has no roots

• The graph has one vertical asymptote $x=-\frac{e}{d}$

  • e.g. the vertical asymptote of $f\left(x\right)=\frac{4x^2+7x-2}{2x+5}$ is $x=-\frac{5}{2}$

• The graph has an oblique asymptote $y=px+q$

  • Which can be found by writing $a{x}^2+bx+c$ in the form $(dx+e)(px+q)+r$

    • Where p, q, r are constants

    • This can be done by polynomial division or comparing coefficients

  • e.g. $4x^2+7x-2$ can be written $\left(2x+5\right)\left(2x-\frac{3}{2}\right)+\frac{11}{2}$

    • the oblique asymptote of $f\left(x\right)=\frac{4x^2+7x-2}{2x+5}$ is $y=2x-\frac{3}{2}$

Linear over quadratic

• For the rational function of the form $f\left(x\right)=\frac{ax+b}{c{x}^2+dx+e}$

  • e.g. $f\left(x\right)=\frac{2x+5}{4x^2+7x-2}$

• The graph has a y-intercept at $(0,\frac{b}{e})$ provided $e\ne 0$

  • e.g. the y-intercept of $f\left(x\right)=\frac{2x+5}{4x^2+7x-2}$ is $\left(0,-\frac{5}{2}\right)$

  • e.g. $f\left(x\right)=\frac{2x+5}{4x^2+7x}$ does not have a y-intercept

• The graph has one root at $x=-\frac{b}{a}$

  • e.g. the root of $f\left(x\right)=\frac{2x+5}{4x^2+7x-2}$ is $\left(-\frac{5}{2},0\right)$

• The graph has can have 0, 1 or 2 vertical asymptotes

  • They are the solutions to $c{x}^2+dx+e=0$

    • e.g. $f\left(x\right)=\frac{2x+5}{4x^2+7x-2}$ has two vertical asymptotes $x=\frac{1}{4}$ and $x=-2$

    • e.g. $f\left(x\right)=\frac{2x+5}{4x^2+4x+1}$ has one vertical asymptote $x=-\frac{1}{2}$

    • e.g. $f\left(x\right)=\frac{2x+5}{4x^2+1}$ has no vertical asymptotes

• The graph has a horizontal asymptote at $y=0$