Part 1 of 5

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Part 1
Identify the​ vertex, the axis of​ symmetry, the maximum or minimum​ value, and the domain and range of the function.
​f(x)equalsminus​(xminus6​)squaredminus20
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Part 1
Identify the vertex.
The coordinates of the vertex are

enter your response here.
​(Type an ordered​ pair.)
The coordinates of the vertex are(Type an ordered pair.)
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Let's solve this step by step. The function given is:

\[
f(x) = - (x - 6)^2 - 20
\]

This is a quadratic function in vertex form, which is generally written as:

\[
f(x) = a(x - h)^2 + k
\]

where:
- \((h, k)\) is the vertex of the parabola.
- \(a\) determines the direction of the parabola (opens upwards if \(a > 0\) and downwards if \(a < 0\)).

### 1. **Identify the vertex:**
In your function:
- \(a = -1\)
- \(h = 6\)
- \(k = -20\)

Thus, the vertex \((h, k)\) is:

\[
\text{Vertex} = (6, -20)
\]

So, the coordinates of the vertex are \((6, -20)\).

### 2. **Axis of symmetry:**
The axis of symmetry for a parabola in vertex form \(f(x) = a(x - h)^2 + k\) is the vertical line \(x = h\). Here, the axis of symmetry is:

\[
x = 6
\]

### 3. **Maximum or Minimum Value:**
- Since \(a = -1\) (which is less than 0), the parabola opens downward, meaning the vertex is a maximum point.
- The maximum value of the function is the \(y\)-coordinate of the vertex, which is \(-20\).

### 4. **Domain:**
The domain of any quadratic function is all real numbers:

\[
\text{Domain} = (-\infty, \infty)
\]

### 5. **Range:**
Since the parabola opens downward and has a maximum value at the vertex \(y = -20\), the range of the function is:

\[
\text{Range} = (-\infty, -20]
\]

### Summary:
- **Vertex:** \((6, -20)\)
- **Axis of Symmetry:** \(x = 6\)
- **Maximum Value:** \(-20\)
- **Domain:** \((-∞, ∞)\)
- **Range:** \((-∞, -20]\)

Would you like more details on any of these steps?

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Here are five related questions you might find useful:
1. How do you convert a standard form quadratic function to vertex form?
2. What is the significance of the vertex in a quadratic function?
3. How do you determine the direction in which a parabola opens?
4. Can a quadratic function have a minimum value if \(a > 0\)?
5. How does the axis of symmetry help in graphing a quadratic function?

**Tip:** Remember, the sign of \(a\) in the vertex form of a quadratic function tells you whether the parabola opens upwards or downwards!

Learn how to identify the vertex, axis of symmetry, maximum value, and domain and range of the quadratic function f(x) = -(x-6)^2 - 20 with a detailed step-by-step solution. Understand the vertex form of quadratic functions, axis of symmetry, and domain and range concepts. Suitable for high school students.

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