Question

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?

Answer 1

No

Explanation:

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.