Quartic graphs

​​In a nutshell

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

Sketching quartic graphs

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

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

Procedure

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

Example 1

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

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

​$$-1 \times -2 \times -3 \times -4 = 24$$​​

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

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

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

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

Identify the coefficient ​of $$x^4$$ by multiplying all the $$x$$ terms together. 

$$x \times x \times x \times x = x^4 = 1x^4$$​​

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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.  ​

Example 2

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

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

​$$-12 \times 3 = -36$$​​

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

Factorise the quadratic equations.

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

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

​$$(x+2)(x+3)(x-6)(2x+1)$$​​

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Find the roots of $$f(x)$$.

$$\begin{aligned} x + 2 &= 0 \\ x &= -2 \\ \\ x+3 &= 0 \\ x &=-3 \\ \\ x-6 &= 0 \\ x &=6 \\ \\ 2x+1 &= 0\\ x&=-\dfrac12\end{aligned}$$​​​

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

Identify the coefficient of $$x^4$$ by multiplying all the $$x$$ terms together.

$$x \times x \times x \times 2x = 2x^4$$​​

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.  

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