How to recognize a parabola from an algebraic expression?

Question

Context. I worked on an intermediate-value-theorem problem that asked whether $f(x) = x^{10} - 10x^2 + 5$ had a root in $(0, 2)$. I solved the problem by computing $f(0)$ and $f(2)$, both of which are positive, making me try again. However, I (incorrectly) thought to myself --- that's a parabola because of the even degree, so let's take a point $x=a$ inside the interval. (Being a parabola, it has to go further down, so I can find whether it gets below the $x$-axis.) Then I tried $f(1)$, which is negative. So, yes, I concluded --- there must be a root by the intermediate value theorem. "Problem solved". But I graphed the function to discover that it's not a parabola.

The question is --- how could I recognize a parabola from the algebraic expression of a function?

Answer 1

All parabolas have a leading term with a "squared" exponent. A polynomial in descending order whose leading coefficient has any exponent other than $2$ is not quadratic.

Answer 2

A parabola where $y$ is a function of $x$ is $y = a x^2 + b x + c$ where $a, b, c$ are constants and $a \ne 0$. A more general parabola is $a x^2 + b x y + c y^2 + d x + e y + f = 0$ where $b^2 - 4 a c = 0$.