01Gquarticgraphs

toc = Quartic Graphs =

A quartic function has degree 4. The general form of a quartic is y = ax 4 +bx 3 +cx 2 + dx + e


 * The sign of a (the coefficient of x 4 ) controls the direction of the curve
 * a > 0, the graph starts in 2nd Quadrant and ends in 1st Quadrant (upright)
 * a < 0, the graph starts in 3rd Quadrant and ends in 4th Quadrant (inverted)
 * The constant term (e) gives the y-intercept
 * A quartic may have zero, one, two, three or four x-intercepts

The Basic Form (Power Form)[[image:01Gquartic2.gif width="264" height="281" align="right" caption="The Basic Quartic"]]
The basic quartic. y = x 4. looks a bit like a parabola but has a wider base and steeper sides.

The basic quartic can be dilated and shifted in the same way as other curves we have studied, producing the power form of the quartic:

... ... y = a(x – h) 4 + k

The usual transformations apply:
 * This quartic has a turning point at (h, k)
 * And is dilated by a factor of a

Factor Form
If a quartic can be factorised,
 * Any linear factors will give the x-intercepts
 * a factor repeated twice indicates a turning point on the x-axis at that x-value
 * a factor repeated three times indicates a stationary point of inflection on the x-axis at that x-value
 * a factor repeated four times indicates a turning point on the x-axis at that x-value
 * (the graph is the basic power quartic shifted sideways)



** Example 1 **


Find the equation of the graph shown here.


 * The graph is clearly a quartic.


 * By observation, the x-intercepts are x = –2, 1, 3


 * The turning point at –2 indicates a repeated factor

... ... ... y = a(x – 1)(x – 3)(x + 2) 2
 * Hence the equation will be:
 * where **a** is a constant


 * Given that the quartic is inverted, we expect ** a ** to be negative.


 * The y-intercept is (0, –3) so substitute (0, –3) into the equation and solve for ** a **

... ... a ( –1 )( –3 )( 2 ) 2 = –3

... ... 12a = –3

,,, ,,, a = – ¼

Hence the equation is:

... ... y = – ¼( x – 1 )( x – 3 )( x + 2 ) 2 Go to top of page flat

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