Find out the line of equation of this parabola.

To find the line of symmetry and the equation of the parabola, follow these steps:

### 1. Line of Symmetry
The line of symmetry (also known as the axis of symmetry) of a parabola given by \( y = ax^2 + bx + c \) is a vertical line that passes through the vertex of the parabola. The equation of this line is given by:
\[ x = -\frac{b}{2a} \]

For the given equation \( y = x^2 - 4x + 6 \):
- \( a = 1 \)
- \( b = -4 \)

So, the line of symmetry is:
\[ x = -\frac{-4}{2(1)} = \frac{4}{2} = 2 \]

Thus, the equation of the line of symmetry is:
\[ x = 2 \]

### 2. Vertex Form of the Parabola
To convert the given equation to vertex form \( y = a(x-h)^2 + k \), where \((h, k)\) is the vertex, we complete the square:

Given: \( y = x^2 - 4x + 6 \)

#### Step 1: Complete the square:
\[ y = x^2 - 4x + 6 \]
\[ y = (x^2 - 4x + 4) - 4 + 6 \]
\[ y = (x - 2)^2 + 2 \]

Thus, the vertex form of the equation is:
\[ y = (x - 2)^2 + 2 \]

### Summary:
- **Equation of the line of symmetry:** \( x = 2 \)
- **Vertex form of the parabola:** \( y = (x - 2)^2 + 2 \)

Would you like any more details or have any questions?

Here are 5 more questions you might want to explore:
1. How do you derive the vertex form of a quadratic equation from the standard form?
2. How can you use the vertex form to find the maximum or minimum value of a quadratic function?
3. How do you determine the direction in which a parabola opens?
4. What is the effect of changing the coefficient \(a\) on the shape of the parabola?
5. How do you solve quadratic equations by factoring?

**Tip:** Converting a quadratic equation to vertex form makes it easier to identify the vertex and the direction of the parabola.

Learn how to find the line of symmetry and the vertex form equation of a parabola using the example equation y = x^2 - 4x + 6. This detailed guide covers key algebraic concepts, including quadratic equations, vertex form, and completing the square. Suitable for high school students and anyone interested in quadratic functions.

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