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We are given the following equation for the height of a projectile, modeled as:

\[
h(t) = at^2 + vt + h_0
\]

Where:
- \( a \) is the acceleration due to gravity, which is \(-16 \, \text{ft/s}^2\),
- \( v \) is the initial velocity (which we will need to solve for),
- \( h_0 \) is the initial height.

From the graph, we can extract key points:
1. At \( t = 0 \), the height \( h(0) = 5 \), meaning \( h_0 = 5 \).
2. The maximum height occurs at \( t = 1 \) when \( h(1) = 21 \).
3. At \( t = 2.15 \), the height returns to zero, meaning \( h(2.15) = 0 \).

### Step 1: Substituting known values into the equation
We know \( a = -16 \), so the height equation becomes:

\[
h(t) = -16t^2 + vt + 5
\]

### Step 2: Using the point at \( t = 1 \) to find \( v \)
At \( t = 1 \), the height is 21, so:

\[
h(1) = -16(1)^2 + v(1) + 5 = 21
\]
\[
-16 + v + 5 = 21
\]
\[
v = 21 + 16 - 5 = 32
\]

So, the initial velocity \( v \) is 32 ft/s.

### Step 3: Writing the equation
The equation modeling the height is now:

\[
h(t) = -16t^2 + 32t + 5
\]

### Step 4: Verifying with the other point
We can check that at \( t = 2.15 \), the height is approximately 0:

\[
h(2.15) = -16(2.15)^2 + 32(2.15) + 5
\]

Calculating this:

\[
h(2.15) = -16(4.6225) + 32(2.15) + 5
\]
\[
h(2.15) = -73.96 + 68.8 + 5 = -0.16
\]

Since this is very close to zero (due to rounding), our equation works.

Thus, the correct equation is:

\[
h(t) = -16t^2 + 32t + 5
\]

Do you want more details or have any questions?

Here are 5 related questions to explore:
1. How is the vertex form of a quadratic equation used to find the maximum height of a projectile?
2. What does the discriminant of a quadratic equation tell us about the motion of the projectile?
3. How can we derive the time of flight for the projectile using this equation?
4. What would happen if the initial velocity were doubled? How would the equation change?
5. How does changing gravity affect the shape of the parabola?

**Tip:** The coefficient of \( t^2 \) in the height equation (which is negative in this case) tells you the direction of the parabola, showing whether the projectile curves downwards or upwards.

The graph below shows the height of a projectile t seconds after it is launched. If acceleration due to gravity is –16 ft/s², which equation models the height of the projectile correctly?

This problem models the height of a projectile using a quadratic equation with an initial velocity and acceleration due to gravity. Learn to calculate the height based on given data points and verify the solution using algebraic methods.

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