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Explainer Video

Tutor: Labib

Summary

​​Cubic graphs

​​In a nutshell

Cubic functions are polynomial functions in the form f(x)=ax3+bx2+cx+df(x) = ax^3 + bx^2 + cx + d f(x)=ax3+bx2+cx+d, where the coefficients are real numbers and a≠0a \ne 0a=0. A graph of a cubic function may have up to three distinct roots where the curve crosses or touches the xxx-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)f(x)f(x): the y-intercepty \text- intercepty-intercept, the roots and the sign of the coefficient of x3x^3x3​.​

f(x)=ax3+bx2+cx+d\boxed{f(x)=ax^3+bx^2+cx+d}f(x)=ax3+bx2+cx+d​​​


Procedure

​1.1.1.​​

Identify the constant, ddd, to find where the curve crosses the yyy​ axis. 

2.2.2.​​

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

​​3.3.3.​​

Factorise the cubic function, f(x)f(x)f(x).​

​4.4.4.​​

Find the roots of f(x)f(x)f(x), which are the values of xxx when f(x)=0f(x)=0f(x)=0​.

5.5.5.​​

Use the roots to mark all the relevant xxx coordinates on the graph. ​

6.6.6.​​

Identify whether the coefficient of x3x^3x3 is positive or negative.

  • For positive coefficients, the graph will start from the bottom left of the 3rd3rd3rd​ quadrant and end in the top right of the 1st1st1st​ quadrant.
  • For negative coefficients, the graph will start from the top left of the 4th4th4th​ quadrant and end in the bottom right of the 2nd2nd2nd​ quadrant.

7.7.7.​​

If you are unable to find the coefficient of x3x^3x3, substitute large positive and negative values of xxx to see whether f(x)f(x)f(x) tends towards ∞\infin∞ or −∞-\infin−∞ at either end of the graph.

8.8.8.​​

Use all the information gathered to sketch the cubic graph. 

​

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


Example 1

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


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

​−1×3×−2=6-1 \times 3 \times -2 = 6−1×3×−2=6​​


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

Find the roots by considering y=0y=0y=0​. 

x−1=0x=1x+3=0x=−3x−2=0x=2\begin{aligned} x-1 &= 0 \\ x &= 1 \\ \\ x+3 &= 0\\ x &=-3 \\ \\ x-2 &= 0 \\ x &=2 \end{aligned}x−1xx+3xx−2x​=0=1=0=−3=0=2​​​


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

Identify the coefficient ​of x3x^3 x3 by multiplying all the xxx terms together. 

x×x×x=x3=1x3x \times x \times x = x^3 = 1x^3x×x×x=x3=1x3​​

​

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 yyy and f(x)f(x)f(x) can be used interchangeably for graphs which only have an xxx and yyy-axis. 

​

Example 2

Factorise and identify the roots of the function, f(x)=x3−8x2+12xf(x) = x^3 - 8x^2 + 12xf(x)=x3−8x2+12x.


Factorise out the common factor of xxx.

​f(x)=x(x2−8x+12)f(x)=x(x^2 - 8x +12)f(x)=x(x2−8x+12)​​


Factorise the quadratic equation using a method of your choice. 

x2−8x+12 x^2 - 8x +12 x2−8x+12

​​

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


x2−6x−2x+12 x^2 - 6x - 2x +12 x2−6x−2x+12​​

x(x−6)−2(x−6)x(x-6) - 2(x-6) x(x−6)−2(x−6)​​

​(x−2)(x−6)(x-2)(x-6)(x−2)(x−6)​​


Therefore, f(x)=x(x−2)(x−6).‾\underline{ f(x) = x(x-2)(x-6).}f(x)=x(x−2)(x−6).​

​

Find the roots of f(x)f(x)f(x)​. ​

​f(x)=0x(x−2)(x−6)=0x=0x−2=0x=2x−6=0x=6\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}f(x)x(x−2)(x−6)xx−2xx−6x​=0=0=0=0=2=0=6​​​


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



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FAQs - Frequently Asked Questions

What do all roots of a function have in common?

They are values of x where f(x)=0.

What is the format of a cubic function?

A cubic function is in the form: ax^3 + bx^2 + cx + d where a is not equal to 0.

How many roots can a cubic graph have?

Cubic functions can have up to three distinct roots.

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