Question

find the vertex and focus of the parabola: x² + 6x - 8y + 1 = 0 vertex = (?, ) focus = (, )

Explanation:

Step1: Rewrite the equation in vertex form

First, complete the square for the \(x\) terms.

$$
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$$

The standard form of a parabola is \((x - h)^{2}=4p(y - k)\), where \((h,k)\) is the vertex.

Step2: Identify the vertex

Comparing \((x + 3)^{2}=8(y + 1)\) with \((x - h)^{2}=4p(y - k)\), we have \(h=-3\), \(k = - 1\). So the vertex \((h,k)=(-3,-1)\)

Step3: Find the value of \(p\)

Since \(4p = 8\), then \(p = 2\)

Step4: Find the focus

The focus of the parabola \((x - h)^{2}=4p(y - k)\) is \((h,k + p)\)
Substitute \(h=-3\), \(k=-1\), \(p = 2\)
The focus is \((-3,-1 + 2)=(-3,1)\)

Answer:

Vertex \(=(-3,-1)\)
Focus \(=(-3,1)\)