Investigating a polynomial

Calculus, grade 10 (5 units) Lesson 13 of 26Learning

Investigating a polynomial

The steps of an investigation

Investigating a function gathers everything a graph tells – without the graph, from the formula. At the end there are enough points and intervals to draw a sketch: a graph whose shape is right, even if it is not exact. The steps:

The domain: a polynomial is defined for every $\displaystyle x$ .
The points on the axes: $\displaystyle {f(x)}={0}$ – on the $\displaystyle x$ -axis; $\displaystyle f(0)$ – on the $\displaystyle y$ -axis.
Even or odd: check whether $\displaystyle {f(-x)}={f(x)}$ or $\displaystyle {f(-x)}={-f(x)}$ . If so, investigating the right-hand side is enough, and the graph is symmetric.
Extremum points: solve $\displaystyle {f'(x)}={0}$ , and make a table of signs of $\displaystyle f'$ .
Where it rises and falls: read it from the same table.
A sketch: mark all the points you found, and join them as the function rises and falls.

Example:

\displaystyle {f(x)}={x^4-4x^3}

The example below investigates $\displaystyle {f(x)}={x^4-4x^3}$ :

The domain: every $\displaystyle x$ .
$\displaystyle {x^4-4x^3}={x^3(x-4)}={0}$ where $\displaystyle {x}={0}$ or $\displaystyle {x}={4}$ : the points on the axes are $\displaystyle (0,0)$ and $\displaystyle (4,0)$ .
$\displaystyle {f(1)}={-3}$ and $\displaystyle {f(-1)}={5}$ : the function is neither even nor odd.
$\displaystyle {f'(x)}={4x^3-12x^2}={4x^2(x-3)}$ , which is 0 at $\displaystyle {x}={0}$ and at $\displaystyle {x}={3}$ .

$\displaystyle x$	$\displaystyle x<0$	$\displaystyle 0$	$\displaystyle 0<x<3$	$\displaystyle 3$	$\displaystyle x>3$
$\displaystyle f'(x)$	−	0	−	0	+
$\displaystyle f(x)$	falls	no extremum	falls	minimum	rises

\displaystyle f(x)=x^{4}-4x^{3}

\displaystyle A=(3,-27)

\displaystyle B=(4,0)

$\displaystyle {f(3)}={81-108}={-27}$ : a minimum point $\displaystyle (3,-27)$ .
At $\displaystyle {x}={0}$ the tangent is horizontal, but the function falls on both sides – there is no extremum there.
The function falls where $\displaystyle x<3$ and rises where $\displaystyle x>3$ .

Investigating in a closed interval

When the function is defined only in a closed interval, such as $\displaystyle -2\leq x\leq4$ , the investigation adds the ends of the interval: they are extremum points too, and candidates for the absolute extremum.

The example below finds the absolute maximum and the absolute minimum of $\displaystyle {f(x)}={x^3-3x^2}$ in the interval $\displaystyle -2\leq x\leq4$ :

$\displaystyle {f'(x)}={3x^2-6x}={3x(x-2)}$ is 0 at $\displaystyle {x}={0}$ and at $\displaystyle {x}={2}$ .
We work out the values at these points and at the ends: $\displaystyle {f(-2)}={-20}$ , $\displaystyle {f(0)}={0}$ , $\displaystyle {f(2)}={-4}$ , $\displaystyle {f(4)}={16}$ .
The greatest value, 16, is the absolute maximum – at an end, $\displaystyle {x}={4}$ . The least, $\displaystyle -20$ , is the absolute minimum – at an end, $\displaystyle {x}={-2}$ .

A parameter

A parameter is a letter standing for a fixed number, like $\displaystyle a$ in $\displaystyle {f(x)}={x^3+ax^2}$ . A fact about the function gives an equation the parameter solves.

The example below finds $\displaystyle a$ , given that $\displaystyle {f(x)}={x^3+ax^2}$ has an extremum point at $\displaystyle {x}={2}$ :

At an extremum point the derivative is 0, so $\displaystyle {f'(2)}={0}$ .
$\displaystyle {f'(x)}={3x^2+2ax}$ , so $\displaystyle {f'(2)}={12+4a}={0}$ , and $\displaystyle {a}={-3}$ .
We check that it really is an extremum: $\displaystyle {f'(x)}={3x^2-6x}={3x(x-2)}$ changes sign at $\displaystyle {x}={2}$ from − to +, so there is a minimum there.

The number of solutions of

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

The solutions of the equation $\displaystyle {f(x)}={k}$ are the $\displaystyle x$ -coordinates of the points where the graph of $\displaystyle f$ meets the horizontal line $\displaystyle {y}={k}$ . So the investigation tells how many solutions there are, even when the equation cannot be solved.

$\displaystyle {f(x)}={x^3-3x}$ has a maximum $\displaystyle (-1,2)$ and a minimum $\displaystyle (1,-2)$ , and it rises without end to the right and falls without end to the left. The board below shows the line $\displaystyle {y}={1}$ :

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

\displaystyle y=1

\displaystyle A=(-1,2)

\displaystyle B=(1,-2)

$\displaystyle -2<k<2$ : the line meets the graph three times – three solutions. So does the equation $\displaystyle {x^3-3x}={1}$ .
$\displaystyle {k}={2}$ or $\displaystyle {k}={-2}$ : the line passes through an extremum point and touches the graph there – two solutions.
$\displaystyle k>2$ or $\displaystyle k<-2$ : one solution.

Exercise 1

The function is $\displaystyle {f(x)}={x^3-3x^2-9x+5}$ .

Find $\displaystyle f'(x)$ .

$\displaystyle f(x)=x^{3}-3x^{2}-9x+5$

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

Work out the $\displaystyle y$ -coordinates of the points you found in part b.

left extremum

$\displaystyle f(x)=x^{3}-3x^{2}-9x+5$

/ – 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

right extremum

$\displaystyle f(x)=x^{3}-3x^{2}-9x+5$

/ – 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

Mark the extremum points on the graph.

\displaystyle f(x)=x^{3}-3x^{2}-9x+5

Write the point on the $\displaystyle y$ -axis, the kind of each extremum point and where the function rises.

Exercise 2

The function is $\displaystyle {f(x)}={x^4-8x^2}$ .

Show that $\displaystyle f$ is even.
What does that say about its graph?

$\displaystyle {f'(x)}={4x^3-16x}$ .
Solve the equation $\displaystyle {f'(x)}={0}$ .

Solve the inequality $\displaystyle f'(x)>0$ : where does $\displaystyle f$ rise?

Mark on the graph the points where it meets the $\displaystyle x$ -axis, and the extremum points.

\displaystyle f(x)=x^{4}-8x^{2}

Exercise 3

The function $\displaystyle {f(x)}={2x^3-3x^2-12x+1}$ is defined in the interval $\displaystyle -2\leq x\leq3$ .
Its derivative is $\displaystyle {f'(x)}={6x^2-6x-12}$ .

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

Work out the values of the function at the ends of the interval and at the points you found.

\displaystyle f(-2)

$\displaystyle f(x)=2x^{3}-3x^{2}-12x+1$

/ – 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

left extremum

$\displaystyle f(x)=2x^{3}-3x^{2}-12x+1$

/ – 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

right extremum

$\displaystyle f(x)=2x^{3}-3x^{2}-12x+1$

/ – 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(3)

$\displaystyle f(x)=2x^{3}-3x^{2}-12x+1$

/ – 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

What is the absolute maximum of $\displaystyle f$ in the interval, and what is the absolute minimum?
At which $\displaystyle x$ is each of them?

maximum

/ – 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 {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

minimum

/ – 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 {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

Exercise 4

The function is $\displaystyle {f(x)}={x^3+ax^2+3}$ ( $\displaystyle a$ is a parameter).
The function has an extremum point at $\displaystyle {x}={-2}$ .

Write an equation for $\displaystyle a$ and solve it.

With the $\displaystyle a$ you found, $\displaystyle {f'(x)}={3x^2+6x}$ .
Solve the equation $\displaystyle {f'(x)}={0}$ .

Mark both extremum points on the graph.

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

Exercise 5

The function is $\displaystyle {f(x)}={x^3-12x}$ .

Mark the extremum points of $\displaystyle f$ on the graph.

\displaystyle f(x)=x^{3}-12x

How many solutions does each equation have?
Explain by the extremum points.

$\displaystyle {x^3-12x}={20}$
$\displaystyle {x^3-12x}={16}$
$\displaystyle {x^3-12x}={0}$
$\displaystyle {x^3-12x}={-10}$

\displaystyle {}={20}

/ – 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 {}={16}

/ – 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 {}={0}

/ – 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 {}={-10}

/ – 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

Formulas: calculus, grade 10 (5 units)

Open as a page

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
$\displaystyle y=b$