Find the vertex and the line of symmetry: y=-2sup(x,2)-4x+6

To find the vertex and the line of symmetry of the quadratic function \( y = -2x^2 - 4x + 6 \), we can follow these steps:

### 1. Identify the coefficients
The quadratic equation is in the form:
\[
y = ax^2 + bx + c
\]
For this equation:
- \( a = -2 \)
- \( b = -4 \)
- \( c = 6 \)

### 2. Calculate the x-coordinate of the vertex
The x-coordinate of the vertex is given by the formula:
\[
x = -\frac{b}{2a}
\]
Substitute the values of \( a \) and \( b \):
\[
x = -\frac{-4}{2(-2)} = \frac{4}{-4} = -1
\]

### 3. Calculate the y-coordinate of the vertex
Substitute \( x = -1 \) back into the original equation to find the y-coordinate:
\[
y = -2(-1)^2 - 4(-1) + 6
\]
\[
y = -2(1) + 4 + 6 = -2 + 4 + 6 = 8
\]
So, the vertex is at \( (-1, 8) \).

### 4. Find the equation of the line of symmetry
The line of symmetry is a vertical line that passes through the vertex. The equation of the line of symmetry is:
\[
x = -1
\]

### Final Answer:
- **Vertex**: \( (-1, 8) \)
- **Line of Symmetry**: \( x = -1 \)

Would you like further details or have any questions?

Here are some related questions:
1. How does the sign of the coefficient \( a \) affect the direction of the parabola?
2. What is the y-intercept of the given quadratic function?
3. How do you find the x-intercepts (roots) of the quadratic function?
4. Can you convert the given quadratic function to its vertex form?
5. What does the vertex represent in the context of a quadratic function?

**Tip:** The vertex form of a quadratic equation can be useful for quickly identifying the vertex and direction of the parabola.

Discover how to find the vertex and line of symmetry of the quadratic function y = -2x^2 - 4x + 6 step-by-step. This problem covers key algebraic concepts, including quadratic equations, vertices of parabolas, and lines of symmetry. Suitable for high school students.

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