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The given equation is:

\[
x = 3y^2 - 6y + 5
\]

To rewrite this equation in vertex form, we need to complete the square for the \(y\)-terms. Here's how to do it step-by-step:

### Step 1: Factor out the coefficient of \(y^2\) from the first two terms:
\[
x = 3(y^2 - 2y) + 5
\]

### Step 2: Complete the square:
To complete the square, take half of the coefficient of \(y\) (which is \(-2\)), square it, and add and subtract this value inside the parentheses.

\[
\left(\frac{-2}{2}\right)^2 = 1
\]

Now, add and subtract 1 inside the parentheses:

\[
x = 3(y^2 - 2y + 1 - 1) + 5
\]

\[
x = 3((y - 1)^2 - 1) + 5
\]

### Step 3: Simplify the expression:
Distribute the 3 and simplify:

\[
x = 3(y - 1)^2 - 3 + 5
\]

\[
x = 3(y - 1)^2 + 2
\]

So, the equation in vertex form is:

\[
x = 3(y - 1)^2 + 2
\]

### Vertex:
The vertex of the parabola is \((2, 1)\). This is the point \((h, k)\) in the vertex form equation \(x = a(y - k)^2 + h\).

### Graphing:
From the given graph, we can see that:
- The vertex is at \((2, 1)\).
- A key point on the graph is \((4, 2)\).

This corresponds with the given equation since the parabola opens to the right (as \(a = 3 > 0\)) and is symmetric around the vertex.

Would you like more details or have any questions?

Here are 5 related questions to explore:
1. How do you complete the square for quadratic equations?
2. What is the general form of a parabola's equation, and how does it relate to its graph?
3. How does the sign of \(a\) in the equation \(x = a(y - k)^2 + h\) affect the direction of the parabola's opening?
4. Can we derive the equation of a parabola if we know its vertex and one additional point?
5. How can we find the axis of symmetry for a parabola?

**Tip:** When graphing parabolas, it's helpful to identify the vertex and another point to accurately plot the curve.

Rewrite the following parabola in vertex form and graph it by moving the vertex and key point.

x = 3y² - 6y + 5

Learn how to rewrite the quadratic equation x = 3y² - 6y + 5 in vertex form by completing the square. Understand how to find the vertex, graph the equation, and analyze key points, suitable for high school algebra students.

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