Vertex Form Calculator

Convert Parabola Equations

Convert the formula of a parabola between standard and vertex forms.

Standard Form: y = ax² + bx + c

x y

How to Convert Between Standard and Vertex Forms

Converting between standard and vertex forms of a parabola equation is a fundamental skill in algebra. It allows us to easily identify key features of the parabola, such as its vertex and direction of opening.

Formulas

Standard Form: \(y = ax^2 + bx + c\)

Vertex Form: \(y = a(x - h)^2 + k\)

Where:

  • \(a\) determines the direction and steepness of the parabola
  • \(b\) and \(c\) are coefficients in the standard form
  • \((h, k)\) is the vertex of the parabola

Calculation Steps

Standard to Vertex Form:

  1. Calculate \(h\): \(h = -\frac{b}{2a}\)
  2. Calculate \(k\): \(k = c - ah^2\)
  3. Write the equation in vertex form: \(y = a(x - h)^2 + k\)

Vertex to Standard Form:

  1. Expand \((x - h)^2\): \(x^2 - 2hx + h^2\)
  2. Distribute \(a\): \(ax^2 - 2ahx + ah^2\)
  3. Add \(k\) to complete the equation
  4. Identify \(b = -2ah\) and \(c = ah^2 + k\)

Example

Let's convert \(y = 2x^2 - 12x + 10\) from standard to vertex form:

  1. Identify \(a = 2\), \(b = -12\), and \(c = 10\)
  2. Calculate \(h\): \[h = -\frac{b}{2a} = -\frac{-12}{2(2)} = 3\]
  3. Calculate \(k\): \[k = c - ah^2 = 10 - 2(3)^2 = 10 - 18 = -8\]
  4. Write the equation in vertex form: \[y = 2(x - 3)^2 - 8\]

Visual Representation

x y 1 2 3 1 2 3 Vertex (3, -8) x = 3 (Axis of Symmetry) Standard Form: y = 2x² - 12x + 10 Vertex Form: y = 2(x - 3)² - 8 Parabola Vertex Point Axis of Symmetry

This diagram illustrates the parabola \(y = 2x^2 - 12x + 10\) or \(y = 2(x - 3)^2 - 8\) in vertex form. The red point (3, -8) is the vertex of the parabola.

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