Let's solve the problem step by step.

The given quadratic function is:
$h(t) = -16t^2 + 64t + 10$
where $h(t)$ represents the height of the roller coaster in meters, and $t$ is the time in seconds.

1. Sketching the graph

To graph this function, you can use Desmos or similar graphing tools by inputting the equation:
$h(t) = -16t^2 + 64t + 10$

• The graph will appear as a downward-opening parabola since the coefficient of $t^2$ is negative ($-16$).

2. Finding the maximum height and time at which it occurs

The maximum height of a parabola occurs at its vertex. For a quadratic equation in standard form $h(t) = at^2 + bt + c$, the time at the vertex is given by:

$t = -\frac{b}{2a}$

Here:

• $a = -16$, $b = 64$, $c = 10$

Substitute into the formula:
$t = -\frac{64}{2(-16)} = -\frac{64}{-32} = 2 \, \text{seconds}$

To find the height at $t = 2$, substitute $t = 2$ into the original equation:
$h(2) = -16(2)^2 + 64(2) + 10$
$h(2) = -16(4) + 128 + 10$
$h(2) = -64 + 128 + 10 = 74 \, \text{meters}$

Answer:

• The maximum height is 74 meters.
• This occurs at 2 seconds.

3. Domain and Range

• Domain: The domain represents the possible values of $t$, which is the time. Since time is non-negative and the parabola starts at $t = 0$, the domain is:
  $t \geq 0 \, \text{(or in interval notation: } [0, \infty))$

• Range: The range represents the possible values of $h(t)$, which is the height. The maximum height is 74 meters, and the height decreases as $t$ moves away from the vertex. Therefore, the range is:
  $h(t) \leq 74 \, \text{(or in interval notation: } (-\infty, 74])$

Final Summary:

1. Graph: Plot $h(t) = -16t^2 + 64t + 10$ in Desmos to see a downward parabola.
2. Maximum height: 74 meters, occurring at $t = 2$ seconds.
3. Domain: $[0, \infty)$; Range: $(-\infty, 74]$.

Do you need further explanation or clarification on any part of the solution?

Related Questions:

1. What happens to the height when $t = 4$ seconds?
2. How can you rewrite $h(t)$ in vertex form?
3. What is the height at $t = 0$ seconds?
4. Find when the roller coaster touches the ground (height = 0).
5. How would the graph change if the $-16$ coefficient were positive?

Tip: For any quadratic equation, the vertex formula $t = -\frac{b}{2a}$ is crucial to finding the maximum or minimum value!