Skill: Recognize quadratic relations from equations

A relation is quadratic if its equation can be written in one of these forms:

$$y=ax^2+bx+c,\qquad a\ne 0$$

or an equivalent form, such as factored form or vertex form. The key feature is that the variable is squared, and the squared term does not disappear when the equation is simplified.

Worked example

Determine whether the relation

$$y=2(x-3)^2+5$$

is quadratic.

Step 1: Look for a squared variable.

The expression contains $(x-3)^2$, so it has a squared term.

Step 2: Rewrite it in standard form.

Expand the square:

$$(x-3)^2=x^2-6x+9$$

Substitute this into the equation:

$$\begin{aligned} y&=2(x^2-6x+9)+5\\ y&=2x^2-12x+18+5\\ y&=2x^2-12x+23 \end{aligned}$$

Step 3: Compare with $y=ax^2+bx+c$.

The equation is now

$$y=2x^2-12x+23$$

so

$$a=2,\qquad b=-12,\qquad c=23.$$

Since $a=2\ne0$, the relation is quadratic.

Remember:

• A squared term with a nonzero coefficient indicates a quadratic relation.
• The values of $b$ or $c$ can be zero.
• If the squared term cancels during simplification, the relation is not quadratic.