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Vertex Form Calculator

Select and type the values to convert quadratic equations between standard and vertex forms.

Standard to Vertex

Vertex to Standard

Vertex Form : f(x) = a (x - h)² + k

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Vertex Form Calculator

Use this vertex form calculator to find the vertex of a quadratic equation and see its graphical representation. You can convert a quadratic equation from its standard form (ax² + bx + c) to vertex form (a(x - h)² + k), or vice versa.

What Is the Vertex Form?

Vertex form is a way to write a quadratic equation that highlights its turning point, called the vertex. For a quadratic function in vertex form, the vertex is at (h, k).

y = a(x - h)² + k

h = x-coordinate of the vertex
k = y-coordinate of the vertex

What Is the Vertex of a Parabola?

The vertex is the point where a parabola intersects its axis of symmetry.

How to Find the Vertex of a Parabola?

The vertex represents either the maximum (parabola opens downward) or minimum (parabola opens upward) value of a quadratic curve.

For a quadratic equation in standard form ax² + bx + c, the vertex coordinates are calculated as:

Vertex Coordinates:

h = -b / 2a

k = c - b² / 4a

Example:

Find the vertex for the parabola:

y = 2(x - (-6))² - 13

Solution:

Vertex form: y = 2(x + 6)² - 13

Standard form: y = 2x² + 24x + 59

Characteristic points:

Vertex: P(-6, -13)
Y-intercept: P(0, 59)

How To Convert Standard Form to Vertex Form

Standard form of a quadratic equation: ax² + bx + c. To convert to vertex form a(x - h)² + k, follow these steps:

Start with the standard form: y = ax² + bx + c
Factor out a from the first two terms: y = a(x² + (b/a)x) + c
Complete the square: add and subtract (b/2a)² inside the parentheses
Simplify: y = a[(x + b/2a)² - (b²/4a²)] + c
Final vertex form: y = a(x - h)² + k, where h = -b/2a and k = c - b²/4a

How To Convert Vertex Form to Standard Form

To convert from vertex form y = a(x - h)² + k to standard form ax² + bx + c:

Expand the square: y = a(x² - 2hx + h²) + k
Distribute a: y = ax² - 2ahx + ah² + k
Compare with standard form: y = ax² + bx + c
Coefficients: b = -2ah, c = ah² + k

Reference:

From Wikipedia: Quadratic function — Etymology, coefficients, variables, bivariate case, forms of a univariate quadratic function, graph of the function.

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