Linear Equation to Standard Form Calculator Parabola

Vertex to Standard Form Conversion:

\[ y = a(x - h)^2 + k \quad \Rightarrow \quad ax^2 + bx + c = y - k \]

Standard Form Equation:

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1. Vertex to Standard Form Conversion

The calculator converts quadratic equations from vertex form (y = a(x - h)² + k) to standard form (ax² + bx + c = y - k). This conversion is useful for analyzing parabola properties and solving quadratic equations.

2. How the Calculator Works

The calculator performs the following steps:

\[ \begin{align*} 1.&\ \text{Expand } (x - h)^2 \text{ to } x^2 - 2hx + h^2 \\ 2.&\ \text{Multiply by } a \text{ to get } ax^2 - 2ahx + ah^2 \\ 3.&\ \text{Add } k \text{ to get } ax^2 - 2ahx + (ah^2 + k) \\ 4.&\ \text{Rearrange to standard form } ax^2 + bx + c = y - k \end{align*} \]

3. Mathematical Explanation

Key Components:

a - Determines parabola's width and direction (up/down)
h - X-coordinate of vertex (axis of symmetry)
k - Y-coordinate of vertex (minimum/maximum point)
b - Linear coefficient in standard form (b = -2ah)
c - Constant term in standard form (c = ah² + k)

4. Using the Calculator

Instructions: Enter the vertex form coefficients a, h, and k. The calculator will output the equivalent standard form equation with all terms expanded.

5. Frequently Asked Questions (FAQ)

Q1: Why convert between forms?
A: Different forms reveal different properties - vertex form shows max/min points, standard form is better for finding roots.

Q2: What if 'a' is negative?
A: The parabola opens downward. The vertex represents the maximum point.

Q3: How does h affect the equation?
A: h shifts the parabola left/right. Positive h moves right, negative h moves left.

Q4: What does k represent?
A: k is the y-value of the vertex and shifts the parabola up/down.

Q5: Can I convert back to vertex form?
A: Yes, through completing the square, but this calculator only handles vertex → standard conversion.