Questions: x^2 + 32x = 4y - 32

Transcript text: x^{2}+32 x=4 y-32

Solution

Solution Steps

Step 1: Identify the Vertex

The vertex of the parabola is directly given by the coordinates \((h, k) = (-16, 8)\).

Step 2: Calculate the Focus

The focus of the parabola is calculated using the formula \((h, k + \frac{1}{4a})\) for \(a > 0\) or \((h, k - \frac{1}{4a})\) for \(a < 0\). Given \(a = -0.125\), the focus is \((h, 10)\).

Step 3: Determine the Directrix

The directrix of the parabola is the line \(y = k - \frac{1}{4a}\) for \(a > 0\) or \(y = k + \frac{1}{4a}\) for \(a < 0\). Given \(a = -0.125\), the directrix is \(y = 6\).

Final Answer:

The vertex is \((-16, 8)\), the focus is \((-16, 10)\), and the directrix is \(y = 6\).

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