Quartic graphs

Topic summary

A quartic equation is one that has \(x^4\) as it’s highest power of x. If you are familiar with quadratic and cubic graphs, a quartic graph is the next local step in the sequence of graphs.

Shape of a quartic graph

\[f(x) = ax^4 + bx^3 + cx^2 + dx + e\]

If \(a > 0\), the graph start and ends at the top.

If \(a < 0\), the graph starts and ends at the bottom.

Intercepts

Y-intercept: The graph intersects the y-axis at \(f(0) = e\).

X-intercepts (Roots): The x-intercepts are the values of \(x\) where \(f(x) = 0\). A quartic function can have zero, one, two, three, or four real roots. To find the roots, solve the quartic equation.

Turning Points

Quartic graphs can have up to three turning points (local maxima and minima), where the direction of the graph changes.

Symmetry

If the cubic and linear terms (\(bx^3\) and \(dx\)) are missing (i.e., \(b = 0\) and \(d = 0\)), the quartic graph has symmetry about the y-axis, resembling a symmetric “W” or “U” shape.