# 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)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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