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The given function is:

\[
h(t) = 16 - t^2
\]

This is a quadratic function in the form \(h(t) = at^2 + bt + c\), where \(a = -1\), \(b = 0\), and \(c = 16\).

### Analysis of the Function:

1. **Vertex:** The function represents a downward-opening parabola (since the coefficient of \(t^2\) is negative).
2. **Maximum Value:** The vertex of the parabola represents the maximum value of the function because it opens downwards.
3. **Vertex Calculation:** The \(t\)-coordinate of the vertex for a parabola \(at^2 + bt + c\) is given by:

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

   Since \(b = 0\) and \(a = -1\), we have:

   \[
   t = -\frac{0}{2(-1)} = 0
   \]

   Substituting \(t = 0\) back into the function to find the maximum height:

   \[
   h(0) = 16 - (0)^2 = 16
   \]

   So, the maximum height (value of \(h(t)\)) is 16 at \(t = 0\).

4. **Intercepts:**
   - **y-intercept:** When \(t = 0\), \(h(0) = 16\).
   - **x-intercepts:** The values of \(t\) for which \(h(t) = 0\):
   
   \[
   16 - t^2 = 0 \implies t^2 = 16 \implies t = \pm 4
   \]

   Thus, the x-intercepts are \(t = 4\) and \(t = -4\).

Would you like more details or have any specific questions?

### Relative Questions to Explore Further:
1. What is the domain and range of the function \(h(t) = 16 - t^2\)?
2. How does the graph of \(h(t) = 16 - t^2\) compare to a standard parabola?
3. How would the function change if the formula were \(h(t) = 16 - 2t^2\)?
4. Can you derive the vertex form of this quadratic function?
5. What does the function \(h(t) = 16 - t^2\) represent in a physical context, like projectile motion?

#### Tip:
Quadratic functions can be analyzed effectively using the vertex form, \(y = a(x - h)^2 + k\), where \((h, k)\) represents the vertex.

What is the maximum height of the function h(t) = 16 - t^2 and where does it occur?

Learn how to find the maximum height of the function h(t) = 16 - t^2, a quadratic equation representing a downward-opening parabola. This problem explores core algebraic concepts such as the vertex form and quadratic functions. Suitable for students in Grades 9-11.

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