Vertex Form

Vertex form makes a parabola's turning point, axis of symmetry, opening, and vertical scale immediately visible.

The vertex anchors a parabola’s position and symmetry. A U-shaped curve is mirrored across a dashed vertical axis. Its highlighted vertex is the turning point, so vertex form and the axis of symmetry can be read from the same picture.

In $y=a(x-h)^2+k$, the numbers tell the story of the parabola before any table is made.

Reading the form

A quadratic in vertex form is

y=a(x-h)^2+k,qquad a\ne0.

Its vertex is $(h,k)$ and its axis of symmetry is $x=h$. The sign inside the brackets is opposite the vertex's $x$-coordinate: $(x-4)^2$ gives $h=4$, while $(x+4)^2=(x-(-4))^2$ gives $h=-4$.

Worked example: read and sketch

Converting from standard form

Completing the square converts $y=ax^2+bx+c$ to vertex form. For

$$ y=x^2+6x+5, $$

add and subtract $9$:

$$ y=(x^2+6x+9)-9+5=(x+3)^2-4. $$

The vertex is $(-3,-4)$. Expanding the final form checks that no value changed.

Finding intercepts

For the $y$-intercept, set $x=0$. For $x$-intercepts, set $y=0$ and isolate the square:

$$ 0=a(x-h)^2+k, $$

(x-h)^2=-\frac{k}{a}.

If $-k/a<0$, there are no real $x$-intercepts. If it is $0$, the vertex touches the axis once. If positive, symmetric roots appear on either side of $h$.

Connecting forms

Standard form is convenient for the $y$-intercept and algebraic operations. Factored form highlights zeros. Vertex form highlights transformations and extrema. Moving between forms is not cosmetic—it reveals different information about the same function.

Common mistakes

Reading $(x-h)$ with the same sign. The vertex coordinate is $h$, so $(x+5)$ means $h=-5$.

Calling $a$ a horizontal stretch. It changes vertical distances and opening.

Forgetting symmetry. Paired points lie equal distances from the axis.

Reporting only the maximum value. In context, the input coordinate may matter just as much.

Ignoring the model's domain. A vertex outside the realistic interval may not be the contextual optimum.