When graphing an equation, are all equations to the nth power that are even parabola's?

So, would #f(x)=4-x^4# be a parabola?

1 Answer
Nov 15, 2016



No. A parabola is a special kind of graph with the property that for some line (the directrix) and a point not on that line (the focus), every point on the graph is equidistant from the directrix and the focus.

Any parabola can be written in the form

#Ax^2+Bxy+Cy^2+Dx+Ey+F = 0#

where #A, B, C, D, E, F# are constants and #B^2=4AC#

Notice that this means any polynomial function with degree #>2# will not have a graph that is a parabola. Even functions can have a shape that is similar to the shape that parabolas have, but will typically be flatter near the vertex. Furthermore, they do not even have to have that shape:


The blue graph is the parabola #y = x^2#

The red graph is #y=x^8#. It has a similar shape to a parabola, but is much flatter near #x=0#.

The green graph is #y=x^4-x^2#. Its shape is clearly distinct from a parabola.