Question

y²=8(x−3)
find the vertex and focus. make sure to enter your answers as coordinates.
vertex:
focus:
write the equation of the directrix, and then sketch the parabola, the directrix and place a dot on the
vertex and the focus.
directrix:

Explanation:

Step1: Recall the standard form of a parabola

The standard form of a parabola of the form \(y^{2}=4p(x - h)\) has vertex \((h,k)\), focus \((h + p,k)\) and directrix \(x=h - p\).
For the given equation \(y^{2}=8(x - 3)\), we can rewrite it as \(y^{2}=4\times2(x - 3)\).

Step2: Identify the vertex

Comparing with the standard form \(y^{2}=4p(x - h)\), here \(h = 3\) and \(k = 0\). So the vertex \((h,k)=(3,0)\).

Step3: Find the value of \(p\)

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

Step4: Calculate the focus

Using the formula for the focus \((h + p,k)\), substituting \(h = 3\), \(p = 2\) and \(k = 0\), we get \((3+2,0)=(5,0)\).

Step5: Determine the directrix

Using the formula for the directrix \(x=h - p\), substituting \(h = 3\) and \(p = 2\), we get \(x=3 - 2=1\).

Answer:

Vertex: \((3,0)\)
Focus: \((5,0)\)
Directrix: \(x = 1\)