Investigating a polynomial

Algebra and calculus, grade 10 (4 units) Lesson 14 of 20Learning

Investigating a polynomial

The steps of an investigation

Investigating a function gathers everything that can be known about it, and a sketch of the graph is drawn from it. For a polynomial the steps are:

The domain – a polynomial is defined for every $\displaystyle x$ .
Even or odd – check whether $\displaystyle {f(-x)}={f(x)}$ (even) or $\displaystyle {f(-x)}={-f(x)}$ (odd); most functions are neither.
The points on the axes – on the $\displaystyle y$ -axis: $\displaystyle {x}={0}$ ; on the $\displaystyle x$ -axis: solve $\displaystyle {f(x)}={0}$ .
Extremum points – solve $\displaystyle {f'(x)}={0}$ , and decide their kind by a table of signs.
Where it increases and decreases – from the same table.
A sketch – mark all the points, and join them as the function increases and decreases.

Example:

\displaystyle {f(x)}={x^3-4x^2+4x}

The example below investigates the function $\displaystyle {f(x)}={x^3-4x^2+4x}$ .

The domain: every $\displaystyle x$ .
Even or odd: $\displaystyle {f(1)}={1}$ and $\displaystyle {f(-1)}={-9}$ . They are neither equal nor opposite, so the function is neither even nor odd.
The points on the axes: $\displaystyle {f(x)}={x(x^2-4x+4)}={x(x-2)^2}$ . It is 0 at $\displaystyle {x}={0}$ and at $\displaystyle {x}={2}$ : the points $\displaystyle (0,0)$ and $\displaystyle (2,0)$ .
The derivative: $\displaystyle {f'(x)}={3x^2-8x+4}={(3x-2)(x-2)}$ , which is 0 at $\displaystyle {x}={\frac{2}{3}}$ and at $\displaystyle {x}={2}$ .

$\displaystyle x$	$\displaystyle x<\frac{2}{3}$	$\displaystyle \frac{2}{3}$	$\displaystyle \frac{2}{3}<x<2$	$\displaystyle 2$	$\displaystyle x>2$
$\displaystyle f'(x)$	+	0	−	0	+
$\displaystyle f(x)$	increases	maximum	decreases	minimum	increases

$\displaystyle {f\left(\frac{2}{3}\right)}={\frac{8}{27}-\frac{16}{9}+\frac{8}{3}}={\frac{32}{27}}$ : a maximum point $\displaystyle \left(\frac{2}{3},\frac{32}{27}\right)$ .
$\displaystyle {f(2)}={0}$ : a minimum point $\displaystyle (2,0)$ – also a point on the $\displaystyle x$ -axis, where the graph touches the axis.

The sketch

The sketch joins the points we found: the graph increases up to the maximum, decreases down to the minimum on the $\displaystyle x$ -axis, and increases again.

\displaystyle f(x)=x^{3}-4x^{2}+4x

\displaystyle A=(0.667,1.185)

\displaystyle B=(2,0)

The number of solutions of

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

The solutions of the equation $\displaystyle {f(x)}={k}$ are the points where the graph meets the horizontal line $\displaystyle {y}={k}$ . The sketch shows how many such points there are for each $\displaystyle k$ :

$\displaystyle k>\frac{32}{27}$ – one solution: the line passes above the maximum.
$\displaystyle {k}={\frac{32}{27}}$ – two solutions: the line touches the graph at the maximum point.
$\displaystyle 0<k<\frac{32}{27}$ – three solutions.
$\displaystyle {k}={0}$ – two solutions: $\displaystyle {x}={0}$ , and $\displaystyle {x}={2}$ , where the line touches the graph at the minimum.
$\displaystyle k<0$ – one solution.

The graph of

\displaystyle |f(x)|

In the graph of $\displaystyle {g(x)}={|f(x)|}$ every negative value of $\displaystyle f$ becomes positive: the part of the graph below the $\displaystyle x$ -axis is reflected above it, and the rest stays.

\displaystyle f(x)=x^{3}-4x^{2}+4x

\displaystyle g(x)=|x^{3}-4x^{2}+4x|

Where $\displaystyle x<0$ the graph of $\displaystyle f$ is below the axis, so the graph of $\displaystyle g$ is its reflection above the axis.
At the origin $\displaystyle g$ has a minimum, $\displaystyle (0,0)$ : the graph comes down to it and goes up from it at a sharp corner, so it has no tangent there.
The maximum $\displaystyle \left(\frac{2}{3},\frac{32}{27}\right)$ and the minimum $\displaystyle (2,0)$ of $\displaystyle f$ are extremum points of $\displaystyle g$ too.

Exercise 1

The function is $\displaystyle {f(x)}={-x^4+2x^3-x^2}$ .

Mark the points where the graph of the function meets the $\displaystyle x$ -axis.

\displaystyle f(x)=-x^{4}+2x^{3}-x^{2}

Find $\displaystyle f'(x)$ .

$\displaystyle f(x)=-x^{4}+2x^{3}-x^{2}$

/ – 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 $\displaystyle {f'(x)}={0}$ .

Mark the extremum points of the function.

\displaystyle f(x)=-x^{4}+2x^{3}-x^{2}

Make a table of signs of $\displaystyle f'(x)$ .
Write where the function increases and decreases, and the kind and the coordinates of each extremum point.

The line is $\displaystyle {y}={k}$ .
For which values of $\displaystyle k$ do the graph of the function and the line meet at 4 points?
At 3 points?
At 2 points?

Exercise 2

The function is $\displaystyle {f(x)}={\frac{1}{3}x^3-2x^2+3x+2}$ .

Find $\displaystyle f'(x)$ .

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

/ – 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 point where the graph meets the $\displaystyle y$ -axis, and the extremum points.

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

Write where the function increases and decreases, and the coordinates and the kind of each extremum point.

How many solutions does the equation $\displaystyle {f(x)}={0}$ have?
Give the reason from the extremum points.

/ – 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

A new function is $\displaystyle {g(x)}={f(x)+k}$ .
Find the $\displaystyle k$ for which the graph of $\displaystyle g$ touches the line $\displaystyle {y}={3}$ at its maximum point.
And find the $\displaystyle k$ for which it touches that line at its minimum point.
Write an equation in $\displaystyle k$ for each case, and solve it.

Exercise 3

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 – a new line · * – times · ^ – power, ↓ – end of the power

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

Write the extremum points of the function, with the coordinates and the kind of each.

Draw a sketch of the graph of the function.

Exercise 4

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

Is the function even, odd or neither?
Give the reason using $\displaystyle f(-x)$ .

Solve the equation $\displaystyle {f(x)}={0}$ : where does the graph meet the $\displaystyle x$ -axis?

Write the equation $\displaystyle {f'(x)}={0}$ and solve it.

Mark the extremum points of the function.

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

How many solutions does the equation $\displaystyle {f(x)}={k}$ have when $\displaystyle {k}={-16}$ ?
When $\displaystyle {k}={-10}$ ?
When $\displaystyle {k}={0}$ ?
When $\displaystyle {k}={5}$ ?
When $\displaystyle {k}={-20}$ ?

\displaystyle {k}={-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 {k}={-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

\displaystyle {k}={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 {k}={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

\displaystyle {k}={-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

Formulas: algebra and calculus, grade 10 (4 units)

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Algebraic technique

The square of a sum
$\displaystyle (a+b)^2$ $\displaystyle =$ $\displaystyle a^2+2ab+b^2$
The square of a difference
$\displaystyle (a-b)^2$ $\displaystyle =$ $\displaystyle a^2-2ab+b^2$
The difference of squares
$\displaystyle (a+b)(a-b)$ $\displaystyle =$ $\displaystyle a^2-b^2$
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 roots of $\displaystyle {ax^2+bx+c}={0}$
$\displaystyle x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
The product of roots
$\displaystyle \sqrt{a}\cdot\sqrt{b}$ $\displaystyle =$ $\displaystyle \sqrt{ab}$ $\displaystyle a\geq0,\ b\geq0$
The quotient of roots
$\displaystyle \frac{\sqrt{a}}{\sqrt{b}}$ $\displaystyle =$ $\displaystyle \sqrt{\frac{a}{b}}$ $\displaystyle a\geq0,\ b>0$

The properties of a function

An even function – its graph symmetric about the $\displaystyle y$ -axis
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle f(x)$
An odd function – its graph symmetric about the origin
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle -f(x)$
The $\displaystyle x$ -coordinate of the vertex of $\displaystyle {y}={ax^2+bx+c}$
$\displaystyle x=\frac{-b}{2a}$
The absolute value
$\displaystyle |a|=\begin{cases}a & a\geq0\\-a & a<0\end{cases}$

Transforming a graph

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

The derivative and the rules of differentiation

The average rate of change from $\displaystyle x_1$ to $\displaystyle x_2$ – the slope of the chord
$\displaystyle \frac{f(x_2)-f(x_1)}{x_2-x_1}$
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 derivative of a product
$\displaystyle (f(x)\cdot g(x))'$ $\displaystyle =$ $\displaystyle f'(x)\cdot g(x)+f(x)\cdot g'(x)$
The derivative of a composite function
$\displaystyle (f(g(x)))'$ $\displaystyle =$ $\displaystyle f'(g(x))\cdot g'(x)$

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
$\displaystyle y-y_1=m(x-x_1)$

The derivative of a square root function

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
$\displaystyle (\sqrt{f(x)})'$ $\displaystyle =$ $\displaystyle \frac{f'(x)}{2\sqrt{f(x)}}$ $\displaystyle f(x)>0$

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
$\displaystyle f'(x)=0$