The function given is:

$f(x) = -(x - 8)^2 + 21$

This is in vertex form, which is:

$f(x) = a(x - h)^2 + k$

where $(h, k)$ represents the vertex.

Identifying the vertex:

From the equation $f(x) = -(x - 8)^2 + 21$, we can compare it with the vertex form and see that:

• $h = 8$
• $k = 21$

Thus, the vertex is $(8, 21)$.

Axis of symmetry:

The axis of symmetry is the vertical line that passes through the vertex, and its equation is $x = h$. Therefore, the axis of symmetry is: $x = 8$

Maximum or minimum value:

Since the coefficient of $(x - 8)^2$ is negative (the equation has a leading $-$), the parabola opens downwards, meaning the vertex is a maximum point.

The maximum value of the function is the $y$-coordinate of the vertex, which is: $\text{Maximum value} = 21$

Domain and range:

• The domain of any quadratic function is all real numbers, so the domain is: $\text{Domain}: (-\infty, \infty)$
• Since the parabola opens downward and has a maximum value of 21, the range is: $\text{Range}: (-\infty, 21]$

Would you like more details or further explanations?

Here are five related questions to practice:

1. How do you determine whether the vertex represents a maximum or minimum?
2. What is the impact of changing the value of $h$ in the vertex form equation?
3. How would the graph change if the coefficient of $(x - 8)^2$ was positive?
4. Can you find the y-intercept of this function?
5. How can you find the x-intercepts of this quadratic function?

Tip: In vertex form, the sign of the coefficient $a$ in $a(x-h)^2 + k$ determines if the parabola opens upwards or downwards.