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!