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 x4) 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)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.
Solution
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
Hence the equation will be:
… … $y = a(x - 1)(x - 3)(x + 2)^2$
… … 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
… … $a(-1)(-3)(2)2 = -3$
… … $12a = -3$
… … $a = -\dfrac{1}{4}$
Hence the equation is:
… … $y = -\dfrac{1}{4}(x - 1)(x - 3)(x + 2)^2$
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