Identifying Characteristics of Quadratic Functions

The general form of a quadratic function is $$\begin{array}{r}y=a{x}^2+bx+c.\end{array}$$ Investigate if the functions are in this form.

Function $A$

The function has two $x^2$ terms. Thus, they can be added. $$\begin{array}{r}y=x^2+3x^2\iff y=4x^2.\end{array}$$ The function does not have an $x$-term or constant but it simply means that these terms are zero, so $$\begin{array}{r}y=4x^2+0x+0.\end{array}$$ Therefore, $A$ is a quadratic function.

Function $B$

The function contains an $x^2$-term, but also the term $x^{0.5}.$ Hence, it does not fit the form $y=a{x}^2+bx+c,$ and is not a quadratic function.

Function $C$

We see directly that the function is not a quadratic function because it contains a variable that is raised to $3.$

Function $D$

The function expression contains a fraction. But, if it can be written in the form $y=a{x}^2+bx+c$, it is still a quadratic function. Investigate whether it is possible to rewrite it. The function $D$ is then a quadratic function.