Cubic graphs

In a nutshell

Cubic functions are polynomial functions in the form $f(x) = ax^3 + bx^2 + cx + d$ , where the coefficients are real numbers and $a \ne 0$ . A graph of a cubic function may have up to three distinct roots where the curve crosses or touches the $x$ -axis.

Sketching cubic graphs

To sketch a cubic graph, there are three important features that need to be identified from a given cubic function, $f(x)$ : the $y \text- intercept$ , the roots and the sign of the coefficient of $x^3$ .

$\boxed{f(x)=ax^3+bx^2+cx+d}$

Procedure

$1.$	Identify the constant, $d$ , to find where the curve crosses the $y$ axis.
$2.$	Mark the $y \text- intercept$ on a graph.
$3.$	Factorise the cubic function, $f(x)$ .
$4.$	Find the roots of $f(x)$ , which are the values of $x$ when $f(x)=0$ .
$5.$	Use the roots to mark all the relevant $x$ coordinates on the graph.
$6.$	Identify whether the coefficient of $x^3$ is positive or negative. For positive coefficients, the graph will start from the bottom left of the $3rd$ quadrant and end in the top right of the $1st$ quadrant. For negative coefficients, the graph will start from the top left of the $4th$ quadrant and end in the bottom right of the $2nd$ quadrant.
$7.$	If you are unable to find the coefficient of $x^3$ , substitute large positive and negative values of $x$ to see whether $f(x)$ tends towards $\infin$ or $-\infin$ at either end of the graph.
$8.$	Use all the information gathered to sketch the cubic graph.

Note: Distinct roots cross the $x$ -axis. Double repeated roots will touch the $x$ -axis. Triple repeated roots will form points of inflection.

Example 1

Sketch the curve for the equation $y= (x-1)(x+3)(x-2)$ .

Identify the $y \text- intercept$ by multiplying all the constant values in the equation. 

$-1 \times 3 \times -2 = 6$

Mark the $y \text- intercept$ on a graph.

Find the roots by considering $y=0$ .

$\begin{aligned} x-1 &= 0 \\ x &= 1 \\ \\ x+3 &= 0\\ x &=-3 \\ \\ x-2 &= 0 \\ x &=2 \end{aligned}$ 

Mark the roots with the $y\text-intercept$ on a graph.

Identify the coefficient of $x^3$ by multiplying all the $x$ terms together.

$x \times x \times x = x^3 = 1x^3$ 

The coefficient is positive, so the curve will start from the bottom left and end in the top right.

Use this information to sketch the curve.

Maths; Sketching graphs; KS5 Year 12; Cubic graphs

Note: The notation $y$ and $f(x)$ can be used interchangeably for graphs which only have an $x$ and $y$ -axis.

Example 2

Factorise and identify the roots of the function, $f(x) = x^3 - 8x^2 + 12x$ .

Factorise out the common factor of $x$ .

$f(x)=x(x^2 - 8x +12)$

Factorise the quadratic equation using a method of your choice.

$x^2 - 8x +12$

$\begin{aligned} 12&= (-6) \times (-2)& \\ -8 &= (-6) + (-2) \end{aligned}$

$x^2 - 6x - 2x +12$

$x(x-6) - 2(x-6)$

$(x-2)(x-6)$

Therefore, $\underline{ f(x) = x(x-2)(x-6).}$

Find the roots of $f(x)$ .

$\begin{aligned} f(x)&=0 \\ x(x-2)(x-6) &=0\\\\ x &= 0 \\ \\ x-2 &=0 \\ x&=2 \\ \\ x-6&=0 \\ x&=6\end{aligned}$

Therefore, the roots of the equation occur when: $\underline{ x=0, 2 \ and \ 6.}$