For the quadratic function f left parenthesis x right parenthesis equals negative 2 x squared plus 2 x minus 6​, answer parts​ (a) through ​(f).
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Part 1
​(a) Find the vertex and the axis of symmetry of the quadratic​ function, and determine whether the graph is concave up or concave down.
The vertex is

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The given quadratic function is:

\[
f(x) = -2x^2 + 2x - 6
\]

### Step 1: Identify whether the graph is concave up or concave down.
The coefficient of \(x^2\) is \(-2\), which is negative. Therefore, the parabola opens **downward**, meaning the graph is concave down.

### Step 2: Find the axis of symmetry.
For a quadratic function in the form \( f(x) = ax^2 + bx + c \), the formula for the axis of symmetry is:

\[
x = \frac{-b}{2a}
\]

Here, \(a = -2\) and \(b = 2\). Substituting these values:

\[
x = \frac{-2}{2(-2)} = \frac{-2}{-4} = \frac{1}{2}
\]

Thus, the axis of symmetry is **\(x = \frac{1}{2}\)**.

### Step 3: Find the vertex.
To find the vertex, substitute \(x = \frac{1}{2}\) into the quadratic function \(f(x)\) to find the corresponding \(y\)-coordinate.

\[
f\left(\frac{1}{2}\right) = -2\left(\frac{1}{2}\right)^2 + 2\left(\frac{1}{2}\right) - 6
\]

Simplifying:

\[
f\left(\frac{1}{2}\right) = -2\left(\frac{1}{4}\right) + 2\left(\frac{1}{2}\right) - 6
\]
\[
= -\frac{2}{4} + 1 - 6 = -\frac{1}{2} + 1 - 6 = \frac{1}{2} - 6 = -\frac{11}{2}
\]

Thus, the vertex is \(\left(\frac{1}{2}, -\frac{11}{2}\right)\).

### Final Answer:
- The vertex is **\(\left(\frac{1}{2}, -\frac{11}{2}\right)\)**.
- The axis of symmetry is **\(x = \frac{1}{2}\)**.
- The graph is **concave down**.

Would you like more details or have any questions? Here are 5 follow-up questions to explore this further:

1. How do you find the y-intercept of a quadratic function?
2. What is the significance of the vertex in a quadratic graph?
3. Can you derive the formula for the vertex from completing the square?
4. How does the sign of the leading coefficient affect the shape of the parabola?
5. What are the properties of the axis of symmetry in any quadratic function?

**Tip:** The vertex form of a quadratic equation can make finding the vertex much easier!

For the quadratic function f(x) = -2x^2 + 2x - 6, answer parts (a) through (f). (a) Find the vertex and the axis of symmetry of the quadratic function, and determine whether the graph is concave up or concave down. The vertex is ________.

Discover how to find the vertex and axis of symmetry for the quadratic function f(x) = -2x^2 + 2x - 6. This detailed step-by-step guide explains key algebraic concepts such as concavity and symmetry. Suitable for students in Grades 8-10.

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