Vertex Form

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

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 $h=-4$.

Opening and vertical scale

The coefficient $a$ controls shape and direction.

• If $a>0$, the parabola opens upward and the vertex is a minimum.
• If $a<0$, it opens downward and the vertex is a maximum.
• If $|a|>1$, it is narrower than $y=x^2$.
• If $0<|a|<1$, it is wider.

The value $a$ multiplies every vertical displacement from the vertex.

Domain, range, and symmetry

Every vertical parabola has domain $x\in\mathbb R$. The range depends on $a$ and $k$:

Points equally far left and right of $x=h$ have the same output. This symmetry can halve the work of making a graph.

Worked example: read and sketch

Converting from standard form

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

add and subtract $9$:

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

Writing an equation from a vertex and point

If the vertex $(h,k)$ and another point are known, substitute the point into vertex form to find $a$.

Suppose the vertex is $(2,-3)$ and the graph passes through $(4,5)$:

Then $8=4a$, so $a=2$. The equation is

Finding intercepts

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

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 $h$.

Modelling maximum and minimum values

Vertex form is valuable when a question asks for an optimum. In a projectile model, a downward-opening parabola's vertex can represent maximum height and the corresponding time. In a revenue model, it may identify maximum revenue within a realistic domain.

Always interpret both coordinates and respect context restrictions; a mathematical parabola continues forever, but a real flight or sales model does not.

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.

Quick self-check

• What are $a$, $h$, and $k$?
• Is the vertex $(h,k)$ and axis $x=h$?
• Does the sign of $a$ match the opening?
• Is the range written with the correct inequality?
• Do intercepts and symmetric points agree with the sketch?

Related topics

• Quadratic Function
• Parabola
• Completing the Square
• Function Transformations