Investigating a rational function

Calculus, grade 10 (5 units) Lesson 23 of 26Learning

Investigating a rational function

The steps of an investigation

Investigating a rational function starts with what the expression itself shows, and only then goes on to the derivative:

The domain: the denominator is not 0.
Asymptotes and removable discontinuities: every zero of the denominator – a vertical asymptote, or a hole if it is a zero of the numerator too; a horizontal asymptote – the value the function approaches as $\displaystyle x$ grows or falls without end.
Even or odd: work out $\displaystyle f(-x)$ and compare it with $\displaystyle f(x)$ and with $\displaystyle -f(x)$ .
The points on the axes: $\displaystyle {f(x)}={0}$ – the numerator is 0; $\displaystyle {x}={0}$ – if 0 is in the domain.
The derivative: by the quotient rule, then solve $\displaystyle {f'(x)}={0}$ .
A table of signs of $\displaystyle f'$ : between any two points where $\displaystyle {f'(x)}={0}$ or where the function is not defined – the sign may change across a vertical asymptote too.
Extremum points, and where it rises and falls – from the table, and then a sketch: the asymptotes first, dashed, then the points you found.

Example:

\displaystyle \frac{x^2}{x^2-4}

The example below investigates $\displaystyle {f(x)}={\frac{x^2}{x^2-4}}$ step by step:

The domain: $\displaystyle {x^2-4}={0}$ where $\displaystyle {x}={\pm2}$ , so $\displaystyle x\neq2$ , $\displaystyle x\neq-2$ .
The numerator is not 0 at $\displaystyle \pm2$ : vertical asymptotes $\displaystyle {x}={2}$ and $\displaystyle {x}={-2}$ .
The numerator and the denominator have the same degree, and the coefficients of $\displaystyle x^2$ are 1 and 1: a horizontal asymptote $\displaystyle {y}={1}$ .
$\displaystyle {f(-x)}={\frac{(-x)^2}{(-x)^2-4}}={f(x)}$ : the function is even, and the graph is symmetric about the $\displaystyle y$ -axis.
The points on the axes: only $\displaystyle (0,0)$ .

The derivative:

f'(x)=\frac{2x(x^2-4)-x^2\cdot2x}{(x^2-4)^2}=\frac{-8x}{(x^2-4)^2}

The denominator is positive throughout the domain, so the sign of $\displaystyle f'$ is the sign of $\displaystyle -8x$ :

$\displaystyle x$	$\displaystyle x<-2$	$\displaystyle -2<x<0$	$\displaystyle 0$	$\displaystyle 0<x<2$	$\displaystyle x>2$
$\displaystyle f'(x)$	+	+	0	−	−
$\displaystyle f(x)$	rises	rises	maximum	falls	falls

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

\displaystyle x=-2

\displaystyle x=2

\displaystyle y=1

\displaystyle (0,0)

The function rises where $\displaystyle x<-2$ and where $\displaystyle -2<x<0$ , and falls where $\displaystyle 0<x<2$ and where $\displaystyle x>2$ .
$\displaystyle (0,0)$ – a maximum point.
Between $\displaystyle -2$ and 2 the function is negative except at 0, so there the graph is a dome with its top at the origin.

A removable discontinuity

When a number is a zero of both the numerator and the denominator, the common factor can be cancelled. The reduced function equals the original one everywhere – except at that point, where the original is not defined. The graph has a hole there: a removable discontinuity.

For example, $\displaystyle {f(x)}={\frac{x^2-x}{x^2-1}}$ :

$\displaystyle {f(x)}={\frac{x(x-1)}{(x-1)(x+1)}}={\frac{x}{x+1}}$ , provided $\displaystyle x\neq1$ .
$\displaystyle {x}={-1}$ is a zero of the denominator only: a vertical asymptote $\displaystyle {x}={-1}$ .
$\displaystyle {x}={1}$ is a zero of both: a hole at the point $\displaystyle {\left(1,\frac{1}{1+1}\right)}={\left(1,\frac{1}{2}\right)}$ .

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

\displaystyle x=-1

\displaystyle y=1

\displaystyle H=(1,0.5)

The derivative is easiest from the reduced expression: $\displaystyle {f'(x)}={\frac{1\cdot(x+1)-x\cdot1}{(x+1)^2}}={\frac{1}{(x+1)^2}>0}$ .
The function rises throughout its domain, and has no extremum points.

Exercise 1

The function is $\displaystyle {f(x)}={\frac{x^2+4}{x}}$ .

What is the domain of $\displaystyle f$ ?
Show that $\displaystyle f$ is odd.
Write its vertical asymptote.

Find $\displaystyle f'(x)$ .

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

/ – 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 $\displaystyle {f'(x)}={0}$ .

Solve the inequality $\displaystyle f'(x)>0$ .

Mark the extremum points on the graph.

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

\displaystyle x=0

Write where the function rises and falls, and the kind of each extremum point.
Notice: the maximum value is less than the minimum value – how can that be?

Exercise 2

The function is $\displaystyle {f(x)}={\frac{x}{(x-1)^2}}$ .

Write on the board the equations of the vertical asymptote and of the horizontal asymptote.

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

Find $\displaystyle f'(x)$ and reduce the fraction.

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

/ – 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 point on the graph.

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

\displaystyle x=1

Make a table of signs of $\displaystyle f'$ – on both sides of $\displaystyle {x}={1}$ too.
Write where the function rises and where it falls.

Exercise 3

The function is $\displaystyle {f(x)}={\frac{x^2-4}{x^2-2x}}$ .

Find the zeros of the denominator.

Reduce the fraction.

/ – 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

Write the domain, the asymptotes and the point where the graph has a hole.

Differentiate the reduced expression.

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

/ – 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

Does $\displaystyle f$ have extremum points?
Write where it rises and where it falls.

Exercise 4

The function is $\displaystyle {f(x)}={\frac{4x}{x^2+1}}$ .

Find $\displaystyle f'(x)$ .

$\displaystyle f(x)=\frac{4x}{x^{2}+1}$

/ – 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 inequality $\displaystyle f'(x)<0$ : where does $\displaystyle f$ fall?

Mark the extremum points on the graph.

\displaystyle f(x)=\frac{4x}{x^{2}+1}

$\displaystyle f$ has the horizontal asymptote $\displaystyle {y}={0}$ .
How many solutions does the equation $\displaystyle {f(x)}={1}$ have? And the equation $\displaystyle {f(x)}={3}$ ?
Explain by the extremum points and the asymptote.

\displaystyle {f(x)}={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

\displaystyle {f(x)}={3}

/ – 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

Exercise 5

Write a rational function that has all these properties:

a vertical asymptote $\displaystyle {x}={2}$ ;
a horizontal asymptote $\displaystyle {y}={3}$ ;
its graph passes through the origin.

Explain why your function fits.

Formulas: calculus, grade 10 (5 units)

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Algebra

The square of a sum and of a difference
$\displaystyle (a\pm b)^2$ $\displaystyle =$ $\displaystyle a^2\pm2ab+b^2$
The difference of two squares
$\displaystyle a^2-b^2$ $\displaystyle =$ $\displaystyle (a-b)(a+b)$
The cube of a sum and of a difference
$\displaystyle (a\pm b)^3$ $\displaystyle =$ $\displaystyle a^3\pm3a^2b+3ab^2\pm b^3$
The sum and the difference of two cubes
$\displaystyle a^3\pm b^3$ $\displaystyle =$ $\displaystyle (a\pm b)(a^2\mp ab+b^2)$
The quadratic formula for $\displaystyle {ax^2+bx+c}={0}$
$\displaystyle x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
Factoring a trinomial – $\displaystyle x_1$ , $\displaystyle x_2$ the roots of $\displaystyle {ax^2+bx+c}={0}$
$\displaystyle ax^2+bx+c$ $\displaystyle =$ $\displaystyle a(x-x_1)(x-x_2)$

The properties of a function

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
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle -f(x)$
The root of a square
$\displaystyle \sqrt{x^2}$ $\displaystyle =$ $\displaystyle |x|$
The absolute value
$\displaystyle |a|=\begin{cases}a & a\geq0\\-a & a<0\end{cases}$

Transformations of a graph

A shift of $\displaystyle k$ units up ( $\displaystyle k<0$ – down)
$\displaystyle y=f(x)+k$
A shift of $\displaystyle p$ units right ( $\displaystyle p<0$ – left)
$\displaystyle y=f(x-p)$
A vertical stretch ( $\displaystyle |a|>1$ ) or squeeze ( $\displaystyle |a|<1$ )
$\displaystyle y=a\cdot f(x)$
A horizontal squeeze by $\displaystyle a$ ( $\displaystyle a>1$ ) or stretch ( $\displaystyle 0<a<1$ )
$\displaystyle y=f(ax)$
A reflection in the $\displaystyle x$ -axis
$\displaystyle y=-f(x)$
A reflection in the $\displaystyle y$ -axis
$\displaystyle y=f(-x)$

The derivative

The average rate of change from $\displaystyle x_1$ to $\displaystyle x_2$ – the slope of the secant
$\displaystyle \frac{f(x_2)-f(x_1)}{x_2-x_1}$
The derivative at a point – the slope of the tangent
$\displaystyle f'(x_0)=\lim_{h\to0}\frac{f(x_0+h)-f(x_0)}{h}$
The derivative of a power
$\displaystyle (x^n)'$ $\displaystyle =$ $\displaystyle nx^{n-1}$
The derivative of a constant
$\displaystyle (c)'$ $\displaystyle =$ $\displaystyle 0$
The derivative of a constant multiple
$\displaystyle (c\cdot f(x))'$ $\displaystyle =$ $\displaystyle c\cdot f'(x)$
The derivative of a sum and of a difference
$\displaystyle (f(x)\pm g(x))'$ $\displaystyle =$ $\displaystyle f'(x)\pm g'(x)$
The instantaneous speed – the derivative of the distance with respect to time
$\displaystyle v(t)$ $\displaystyle =$ $\displaystyle s'(t)$

Rules of differentiation

The derivative of a product of functions
$\displaystyle [f(x)\cdot g(x)]'$ $\displaystyle =$ $\displaystyle f'(x)\cdot g(x)+f(x)\cdot g'(x)$
The derivative of a quotient of functions
$\displaystyle \left[\frac{f(x)}{g(x)}\right]'$ $\displaystyle =$ $\displaystyle \frac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{[g(x)]^2}$
The derivative of a composite function – the outer derivative times the inner derivative
$\displaystyle [f(u(x))]'$ $\displaystyle =$ $\displaystyle f'(u)\cdot u'(x)$
The derivative of $\displaystyle \frac{1}{x}$
$\displaystyle \left(\frac{1}{x}\right)'$ $\displaystyle =$ $\displaystyle -\frac{1}{x^2}$
The derivative of $\displaystyle \sqrt{x}$
$\displaystyle (\sqrt{x})'$ $\displaystyle =$ $\displaystyle \frac{1}{2\sqrt{x}}$ $\displaystyle x>0$
The derivative of the root of a function
$\displaystyle (\sqrt{f(x)})'$ $\displaystyle =$ $\displaystyle \frac{f'(x)}{2\sqrt{f(x)}}$ $\displaystyle f(x)>0$

The tangent to a graph

The slope of the tangent at the point $\displaystyle (x_1,y_1)$ on the graph
$\displaystyle m=f'(x_1)$
The equation of the tangent
$\displaystyle y-y_1=m(x-x_1)$

Rising, falling and extremum points

The function rises
$\displaystyle f'(x)>0$
The function falls
$\displaystyle f'(x)<0$
A point that may be an extremum
$\displaystyle f'(x)=0$

Asymptotes

A vertical asymptote – $\displaystyle a$ a zero of the denominator that is not a zero of the numerator
$\displaystyle x=a$
A horizontal asymptote – $\displaystyle b$ the value the function approaches as $\displaystyle x$ grows or falls without end
$\displaystyle y=b$