Result

Calculation Steps

Visual Representation

About Parabolas

What is a Parabola?

A parabola is a U-shaped curve that represents a quadratic function. It's defined by the general equation \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a \neq 0\).

Formula

The standard form of a parabola is:

\[f(x) = a(x - h)^2 + k\]

Where:

• \(a\) determines the direction and width of the parabola
• \((h, k)\) is the vertex of the parabola

Key Points of a Parabola

1. Vertex: \((h, k) = (-\frac{b}{2a}, f(-\frac{b}{2a}))\)
2. Axis of Symmetry: \(x = -\frac{b}{2a}\)
3. Focus: \((h, k + \frac{1}{4a})\) for \(a > 0\), or \((h, k - \frac{1}{4a})\) for \(a < 0\)
4. Directrix: \(y = k - \frac{1}{4a}\) for \(a > 0\), or \(y = k + \frac{1}{4a}\) for \(a < 0\)

Calculation Steps

1. Identify \(a\), \(b\), and \(c\) from the given equation
2. Calculate the x-coordinate of the vertex: \(h = -\frac{b}{2a}\)
3. Calculate the y-coordinate of the vertex: \(k = f(h) = a(h)^2 + bh + c\)
4. Determine the focus and directrix
5. Find x-intercepts (if any) using the quadratic formula
6. Find the y-intercept by calculating \(f(0)\)

Example

Let's analyze the parabola \(f(x) = 2x^2 - 4x - 2\):

1. \(a = 2\), \(b = -4\), \(c = -2\)
2. Vertex x-coordinate: \(h = -\frac{b}{2a} = -\frac{-4}{2(2)} = 1\)
3. Vertex y-coordinate: \(k = f(1) = 2(1)^2 - 4(1) - 2 = -4\)
4. Vertex: (1, -4)
5. Focus: \((1, -4 + \frac{1}{4(2)}) = (1, -3.875)\)
6. Directrix: \(y = -4 - \frac{1}{4(2)} = -4.125\)