Investigating a rational function

Calculus, grade 11 (4 units) Lesson 8 of 20Learning

Investigating a rational function

The steps of an investigation

Investigating a function is a fixed list of questions whose answers together give the shape of the graph:

the domain – every $\displaystyle x$ except the zeros of the denominator;
asymptotes parallel to the axes – vertical ones at the points outside the domain, and a horizontal one as $\displaystyle x\to\pm\infty$ ;
the points on the axes – on the $\displaystyle x$ -axis: $\displaystyle {f(x)}={0}$ ; on the $\displaystyle y$ -axis: $\displaystyle f(0)$ , if 0 is in the domain;
the derivative $\displaystyle f'(x)$ , and the solutions of $\displaystyle {f'(x)}={0}$ ;
the sign table of $\displaystyle f'$ – from it, where the function rises and falls, and the kind of each extremum point;
a sketch of the graph.

The sign table of

\displaystyle f'

$\displaystyle f'$ can change its sign only at a point where $\displaystyle {f'(x)}={0}$ , or at a point where the function is not defined. So:

mark both kinds of point on the $\displaystyle x$ -axis, in order;
on each interval between them, substitute one number in $\displaystyle f'$ and see its sign;
$\displaystyle f'>0$ – the function rises; $\displaystyle f'<0$ – the function falls;
at a point where $\displaystyle {f'}={0}$ and the sign changes from $\displaystyle +$ to $\displaystyle -$ there is a maximum, and from $\displaystyle -$ to $\displaystyle +$ – a minimum;
at a point outside the domain there is no extremum, even when the sign changes there: the graph has no point there.

Example

$\displaystyle {f(x)}={\frac{x^{2}}{x-1}}$ .
In the investigation below we go through the steps in order:

The domain: $\displaystyle x\neq1$ .
Asymptotes: $\displaystyle {x}={1}$ – vertical. As $\displaystyle x\to\pm\infty$ , the numerator grows faster than the denominator, and $\displaystyle f$ comes close to no number: there is no horizontal asymptote.
Points on the axes: $\displaystyle {f(x)}={0}$ only when $\displaystyle {x}={0}$ – the point $\displaystyle (0,0)$ .
$\displaystyle {f'(x)}={\frac{x^{2}-2x}{(x-1)^{2}}}={\frac{x(x-2)}{(x-1)^{2}}}$ , and $\displaystyle {f'(x)}={0}$ when $\displaystyle {x}={0}$ or $\displaystyle {x}={2}$ .
The sign table – the points $\displaystyle 0$ , $\displaystyle 1$ (outside the domain) and $\displaystyle 2$ :

$\displaystyle x$	$\displaystyle x<0$	$\displaystyle 0$	$\displaystyle 0<x<1$	$\displaystyle 1$	$\displaystyle 1<x<2$	$\displaystyle 2$	$\displaystyle x>2$
$\displaystyle f'(x)$	$\displaystyle +$	$\displaystyle 0$	$\displaystyle -$	none	$\displaystyle -$	$\displaystyle 0$	$\displaystyle +$
$\displaystyle f(x)$	rises	maximum	falls	none	falls	minimum	rises

$\displaystyle {f(0)}={0}$ and $\displaystyle {f(2)}={\frac{4}{1}}={4}$ : a maximum $\displaystyle (0,0)$ and a minimum $\displaystyle (2,4)$ .

\displaystyle f(x)=\frac{x^{2}}{x-1}

\displaystyle x=1

\displaystyle (0,0)

\displaystyle (2,4)

The minimum $\displaystyle (2,4)$ is higher than the maximum $\displaystyle (0,0)$ : they are on two branches of the graph, on the two sides of the asymptote $\displaystyle {x}={1}$ .

Exercise 1

$\displaystyle {f(x)}={\frac{-x^{2}+2x+3}{x^{2}}}$ .

Find the domain, and write the equations of the asymptotes of $\displaystyle f$ parallel to the axes on the board.

\displaystyle f(x)=\frac{-x^{2}+2x+3}{x^{2}}

Find the points where the graph meets the axes.

Find $\displaystyle f'(x)$ .

$\displaystyle f(x)=\frac{-x^{2}+2x+3}{x^{2}}$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Solve the equation $\displaystyle {f'(x)}={0}$ .

Write the sign table of $\displaystyle f'$ , and where $\displaystyle f$ rises and falls.

Mark the extremum point of $\displaystyle f$ . What kind is it?

\displaystyle f(x)=\frac{-x^{2}+2x+3}{x^{2}}

\displaystyle x=0

\displaystyle y=-1

Exercise 2

$\displaystyle {f(x)}={\frac{x}{x^{2}+1}}$ .

Find $\displaystyle f'(x)$ .

$\displaystyle f(x)=\frac{x}{x^{2}+1}$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Solve the equation $\displaystyle {f'(x)}={0}$ .

Mark the extremum points of $\displaystyle f$ .

\displaystyle f(x)=\frac{x}{x^{2}+1}

$\displaystyle f$ has no vertical asymptote. Why? What is its horizontal asymptote?

Exercise 3

$\displaystyle {f(x)}={\frac{4}{x^{2}-4}}$ .
Find $\displaystyle f'(x)$ .

$\displaystyle f(x)=\frac{4}{x^{2}-4}$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

The sign table of $\displaystyle f'$ has three points: two points outside the domain and one point where $\displaystyle {f'(x)}={0}$ . Write the table, where $\displaystyle f$ rises and falls, and its extremum point.

Formulas: calculus, grade 11 (4 units)

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Transformations of the graph of $f$

A horizontal shift: $\displaystyle p$ units to the right ( $\displaystyle p>0$ ) or to the left ( $\displaystyle p<0$ )
$\displaystyle y=f(x-p)$
A vertical shift: $\displaystyle k$ units up ( $\displaystyle k>0$ ) or down ( $\displaystyle k<0$ )
$\displaystyle y=f(x)+k$
A reflection in the $\displaystyle x$ -axis
$\displaystyle y=-f(x)$
A reflection in the $\displaystyle y$ -axis
$\displaystyle y=f(-x)$
A vertical stretch ( $\displaystyle a>1$ ) or squeeze ( $\displaystyle 0<a<1$ )
$\displaystyle y=a\cdot f(x)$
Absolute value: the part below the $\displaystyle x$ -axis is reflected above it
$\displaystyle y=|f(x)|$

The rational function

The horizontal asymptote when the numerator and the denominator have the same degree – $\displaystyle a$ and $\displaystyle b$ the coefficients of the highest power
$\displaystyle y=\frac{a}{b}$
The horizontal asymptote when the numerator’s degree is lower than the denominator’s
$\displaystyle y=0$
An even function – the graph is symmetric about the $\displaystyle y$ -axis
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle f(x)$
An odd function – the graph is symmetric about the origin
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle -f(x)$

Derivatives and the tangent

The derivative of a power
$\displaystyle \left(x^{n}\right)'=nx^{n-1}$
The derivative of a product
$\displaystyle \left(u\cdot v\right)'$ $\displaystyle =$ $\displaystyle u'\cdot v+u\cdot v'$
The derivative of a quotient
$\displaystyle \left(\frac{u}{v}\right)'$ $\displaystyle =$ $\displaystyle \frac{u'\cdot v-u\cdot v'}{v^{2}}$
The derivative of $\displaystyle \frac{c}{x}$
$\displaystyle \left(\frac{c}{x}\right)'$ $\displaystyle =$ $\displaystyle -\frac{c}{x^{2}}$
The derivative of a root – $\displaystyle u$ a function of $\displaystyle x$
$\displaystyle \left(\sqrt{u}\right)'$ $\displaystyle =$ $\displaystyle \frac{u'}{2\sqrt{u}}$
The equation of the tangent at the point $\displaystyle (x_1,y_1)$
$\displaystyle y-y_1$ $\displaystyle =$ $\displaystyle m(x-x_1)$
The slope of the tangent
$\displaystyle m$ $\displaystyle =$ $\displaystyle f'(x_1)$

Investigating a function with its derivative

The function rises
$\displaystyle f'(x)>0$
The function falls
$\displaystyle f'(x)<0$
A possible extremum point – check the sign of $\displaystyle f'$ on each side of it
$\displaystyle f'(x)=0$

Integrals and areas

The integral of a power
$\displaystyle \int x^{n}\,dx$ $\displaystyle =$ $\displaystyle \frac{x^{n+1}}{n+1}+C$ $\displaystyle n\neq-1$
The integral of $\displaystyle \frac{c}{(ax+b)^{2}}$
$\displaystyle \int\frac{c}{(ax+b)^{2}}\,dx$ $\displaystyle =$ $\displaystyle -\frac{c}{a(ax+b)}+C$
The definite integral – $\displaystyle F$ an antiderivative of $\displaystyle f$
$\displaystyle \int_{a}^{b}f(x)\,dx$ $\displaystyle =$ $\displaystyle F(b)-F(a)$
The area between two graphs, from $\displaystyle {x}={a}$ to $\displaystyle {x}={b}$
$\displaystyle S$ $\displaystyle =$ $\displaystyle \int_{a}^{b}\left(f(x)-g(x)\right)dx$ $\displaystyle f(x)\geq g(x)$