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

## So, would $f \left(x\right) = 4 - {x}^{4}$ be a parabola?

Nov 15, 2016

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

$A {x}^{2} + B x y + C {y}^{2} + D x + E y + F = 0$

where $A , B , C , D , E , F$ are constants and ${B}^{2} = 4 A C$

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.