Question

y² - 6y + 3x - 3 = 0
find the vertex.
(x, y)
find the focus.
(x, y)
find an equation for the directrix.
graph the parabola using the focus and directrix.

Explanation:

Step1: Rewrite the equation in standard form

Complete the square for the \(y\) - terms.

$$
LATEXBLOCK0
$$

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

Step2: Find the vertex

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

Step3: Find the value of \(p\)

Since \(4p=-3\), then \(p=-\frac{3}{4}\)

Step4: Find the focus

The focus of the parabola \((y - k)^{2}=4p(x - h)\) is \((h + p,k)\)
Substitute \(h = 4\), \(k = 3\) and \(p=-\frac{3}{4}\)
\(h+p=4-\frac{3}{4}=\frac{16 - 3}{4}=\frac{13}{4}\)
So the focus \((x,y)=(\frac{13}{4},3)\)

Step5: Find the equation of the directrix

The equation of the directrix of the parabola \((y - k)^{2}=4p(x - h)\) is \(x=h - p\)
Substitute \(h = 4\) and \(p=-\frac{3}{4}\)
\(x=4-(-\frac{3}{4})=4+\frac{3}{4}=\frac{16 + 3}{4}=\frac{19}{4}\)

Answer:

• Vertex: \((4,3)\)
• Focus: \((\frac{13}{4},3)\)
• Directrix: \(x = \frac{19}{4}\)