How to find the axis of symmetry of a quadratic from standard form

Key takeaway: For a quadratic in standard form, the axis of symmetry is usually the fastest thing to extract. You do not need to complete the square if the question only asks for the symmetry line.

For

$y=3x^2+12x+40,$

we identify

• $a=3$
• $b=12$

The axis of symmetry of $y=ax^2+bx+c$ is

$x=-\frac{b}{2a}$

Substitute the values:

$x=-\frac{12}{2(3)}=-\frac{12}{6}=-2$

So the axis of symmetry is

$\boxed{x=-2}$

If you want to check it another way, the vertex must lie halfway between any pair of symmetric points. The formula gives that midpoint directly, which is why it is the standard shortcut in exams.

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Pitfalls the pros know 👇 Students sometimes write $y=-2$ instead of $x=-2$. The axis of symmetry is a vertical line, so the answer must be an equation in $x$, not a coordinate pair. Another easy mistake is using the wrong sign in $-\frac{b}{2a}$.

What if the problem changes? If the quadratic were written as $y=3(x+2)^2+\text{constant}$, the axis of symmetry would still be $x=-2$. If the coefficient of $x^2$ changed, you would still use the same formula; only the final value would differ.

Tags: axis of symmetry, vertex, quadratic form