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Quartic graphs

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Explainer Video

Tutor: Labib

Summary

Quartic graphs

​​In a nutshell

Quartic functions are polynomials in the form f(x)=ax4+bx3+cx2+dx+ef(x) = ax^4 + bx^3 + cx^2 + dx + e f(x)=ax4+bx3+cx2+dx+e. The coefficients are real numbers and a≠0a \ne 0a=0. A graph of a quartic function can have up to four distinct roots where the curve crosses or touches the xxx axis. ​



Sketching quartic graphs

To sketch a quartic graph, there are three important features that need to be identified from a given quartic function, f(x)f(x)f(x): the y-intercepty \text- intercepty-intercept, the roots and the sign of the coefficient of x4x^4x4​.​

​f(x)=ax4+bx3+cx2+dx+e\boxed{f(x)=ax^4+bx^3+cx^2+dx+e}f(x)=ax4+bx3+cx2+dx+e​​​


Procedure

​1.1.1.​​

Identify the constant, eee, 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 quartic 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 x4x^4x4 is positive or negative.

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

7.7.7.​​

If you are unable to find the coefficient of x4x^4x4, 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 quartic graph. 

​

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


Example 1

Sketch the curve for the equation y=(x−1)(x−2)(x−3)(x−4)y= (x-1)(x-2)(x-3)(x-4)y=(x−1)(x−2)(x−3)(x−4).


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

​−1×−2×−3×−4=24-1 \times -2 \times -3 \times -4 = 24−1×−2×−3×−4=24​​


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

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

x−1=0x=1x−2=0x=2x−3=0x=3x−4=0x=4\begin{aligned} x-1 &= 0 \\ x &= 1 \\ \\ x-2 &= 0\\ x &=2 \\ \\ x-3 &= 0 \\ x &=3 \\ \\ x-4 &= 0\\ x&=4\end{aligned}x−1xx−2xx−3xx−4x​=0=1=0=2=0=3=0=4​​​


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

Identify the coefficient ​of x4x^4x4 by multiplying all the xxx terms together. 

x×x×x×x=x4=1x4x \times x \times x \times x = x^4 = 1x^4x×x×x×x=x4=1x4​​

​

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

Use the gathered information to sketch the curve.  ​

Maths; Sketching graphs; KS5 Year 12; Quartic graphs


Example 2

Sketch the function, f(x)=(x2−4x−12)(2x2+7x+3)f(x) = (x^2-4x-12)(2x^2+7x+3)f(x)=(x2−4x−12)(2x2+7x+3).


Identify the y-intercepty\text-intercepty-intercept by multiplying the constant values.

​−12×3=−36-12 \times 3 = -36−12×3=−36​​


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

Factorise the quadratic equations.

x2−4x−12=(x−6)(x+2)2x2+7x+3=(2x+1)(x+3)\begin{aligned} x^2 - 4x -12 &= (x-6)(x+2) \\ 2x^2 +7x + 3 &= (2x+1)(x+3) \end{aligned}x2−4x−122x2+7x+3​=(x−6)(x+2)=(2x+1)(x+3)​

​​​

Rewrite the fully factorised form of f(x)f(x)f(x).​

​(x+2)(x+3)(x−6)(2x+1)(x+2)(x+3)(x-6)(2x+1)(x+2)(x+3)(x−6)(2x+1)​​

​​

Find the roots of f(x)f(x)f(x).

x+2=0x=−2x+3=0x=−3x−6=0x=62x+1=0x=−12\begin{aligned} x + 2 &= 0 \\ x &= -2 \\ \\ x+3 &= 0 \\ x &=-3 \\ \\ x-6 &= 0 \\ x &=6 \\ \\ 2x+1 &= 0\\ x&=-\dfrac12\end{aligned}x+2xx+3xx−6x2x+1x​=0=−2=0=−3=0=6=0=−21​​​​​


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

Identify the coefficient of x4x^4x4 by multiplying all the xxx terms together.

x×x×x×2x=2x4x \times x \times x \times 2x = 2x^4x×x×x×2x=2x4​​


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

Use the gathered information to sketch the curve.  

Maths; Sketching graphs; KS5 Year 12; Quartic graphs

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FAQs - Frequently Asked Questions

How many roots can a quartic graph have?

A quartic graph can have up to 4 distinct roots.

What is the format of a quartic function?

A quartic function is in the form: ax^4+bx^3+cx^2+dx+e where a is not equal to 0.

How do you sketch quadruple repeated roots?

Quadruple repeated roots are points where a curve only touches the x axis and does not cross it.

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