calculation focus

Vertex Form Calculator

Convert any quadratic equation from standard form (ax²+bx+c) to vertex form a(x-h)²+k. Find the vertex, axis of symmetry, and graph the parabola.

Back to Quadratic Formula Calculator

Quick Tips

Click to show tips

Try an Example

Pick a scenario to see how the calculator works, then adjust the values

Standard Form

Classic quadratic x² − 5x + 6 = 0 with two integer roots (x = 2 and x = 3)

Key values: a = 1 · b = −5 · c = 6

Physics Problem

Projectile height equation −4.9t² + 20t + 1.5 = 0 (when does the ball hit the ground?)

Key values: a = −4.9 · b = 20 · c = 1.5

Complex Roots

Equation x² + 2x + 5 = 0 with no real solutions (discriminant < 0)

Key values: a = 1 · b = 2 · c = 5

Documentation

Vertex Form of a Quadratic

The vertex form writes a quadratic so the vertex is immediately visible:

y = a(x - h)^2 + k

where $(h, k)$ is the vertex (the highest or lowest point of the parabola) and $a$ controls the width and direction.

Converting from Standard Form

Given $y = ax^2 + bx + c$ , the vertex coordinates are:

h = -\frac{b}{2a} \qquad k = c - \frac{b^2}{4a}

Example: $y = 2x^2 - 12x + 22$

$h = \frac{12}{4} = 3$ , $k = 22 - \frac{144}{8} = 22 - 18 = 4$

Vertex form: $y = 2(x - 3)^2 + 4$ , vertex at (3, 4)

What Each Parameter Controls

Parameter	Effect	Example
$a > 0$	Opens upward (minimum at vertex)	$y = (x-1)^2 + 2$
$a < 0$	Opens downward (maximum at vertex)	$y = -(x-1)^2 + 2$
$\|a\| > 1$	Narrower than standard parabola	$y = 3x^2$ (steep)
$\|a\| < 1$	Wider than standard parabola	$y = 0.5x^2$ (shallow)
$h$	Horizontal shift (right if positive)	$(x - 3)^2$ shifts right 3
$k$	Vertical shift (up if positive)	$x^2 + 5$ shifts up 5

Vertex Form for Optimization

Vertex form instantly reveals the maximum or minimum value of a quadratic function — it's $k$ , occurring at $x = h$ .

Projectile example: If a ball's height is $h(t) = -5(t - 3)^2 + 45$ , the maximum height is 45 meters at time t = 3 seconds. No calculus needed.

Frequently Asked Questions

What is vertex form of a quadratic equation?

Vertex form is $y = a(x - h)^2 + k$ , where $(h, k)$ is the vertex of the parabola. It makes the vertex coordinates immediately visible, unlike standard form $y = ax^2 + bx + c$ where you need to calculate $h = -b/(2a)$ and $k = c - b^2/(4a)$ .

How do I convert from standard form to vertex form?

Use the formulas $h = -b/(2a)$ and $k = c - b^2/(4a)$ . For example, $y = 2x^2 - 12x + 22$ gives $h = 12/4 = 3$ , $k = 22 - 144/8 = 4$ , so vertex form is $y = 2(x - 3)^2 + 4$ .

What does the parameter $a$ control in vertex form?

The parameter $a$ controls the width and direction of the parabola. If $a > 0$ the parabola opens upward (minimum at vertex); if $a < 0$ it opens downward (maximum at vertex). Larger $|a|$ makes the parabola narrower, smaller $|a|$ makes it wider.

How does vertex form help with optimization problems?

Vertex form instantly reveals the maximum or minimum value of the function: it is $k$ , occurring at $x = h$ . For example, if a ball height is $h(t) = -5(t - 3)^2 + 45$ , the maximum height is 45 meters at $t = 3$ seconds, with no calculus required.

What is the relationship between vertex form and completing the square?

Completing the square is the algebraic process that transforms standard form into vertex form. You create a perfect square trinomial by adding and subtracting $(b/2a)^2$ , then factor to get $a(x - h)^2 + k$ . The two forms express the same function differently.

Related calculation-focus Variants

Explore more calculation-focus options

discriminant

Discriminant Calculator

Use this calculator

More Math Calculators

Explore the category

Absolute Value Calculator

Area Calculator

Circle Calculator

Percentage Calculator

Prime Factorization Calculator

Unit Circle Explorer