Question

interpret the following quadratic function model and graph given the following context. A volleyball is served into the air at 26 ft./sec
26

ft./sec
from a height of 4.5 ft. The quadratic equation represents the height of the ball over time and in seconds. The graph illustrates this path where x represents the time in seconds and f(x)
represents the height in feet. Approximately how long does it take for the volleyball to reach maximum height?

Answer 1

To interpret the quadratic function model and graph for the volleyball's trajectory, we need to consider the general form of a quadratic function, which is typically expressed as:

\[ f(x) = ax^2 + bx + c \]

In your context, \( f(x) \) represents the height of the volleyball in feet, where \( x \) is the time in seconds.

### Given Information:
- The initial height from which the volleyball is served is 4.5 feet (this is often the constant term \( c \) in the equation).
- The initial velocity (or vertical speed) of the volleyball when served is 26 ft/sec.

### Shape of the Graph:
The graph of a quadratic function is a parabolic curve. Since the volleyball is thrown upward, it will initially rise and then eventually fall back down, forming an "upwards opening" parabola.

### Finding Maximum Height:
The maximum height occurs at the vertex of the parabola. For a quadratic equation in standard form \( ax^2 + bx + c \), the x-coordinate of the vertex (which gives the time at which maximum height is reached) can be calculated using the formula:

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

Where:
- \( a \) is the coefficient of the \( x^2 \) term (which indicates the direction of the parabola),
- \( b \) is the coefficient of the \( x \) term (which influences the position of the vertex).

### Time to Reach Maximum Height:
1. **Identify Coefficients**: From the context, you would need the specific values of \( a \) and \( b \) from the quadratic equation to apply the formula. If the equation is provided, simply substitute \( a \) and \( b \) into the formula above.

2. **Substituting Values**: Once you have those values, you can calculate the time \( x \) when the volleyball reaches its maximum height.

### Conclusion:
- The point at which the volleyball reaches maximum height corresponds to the vertex of the parabola.
- Analyze the provided graph (if available) to find the vertex or apply the formula to determine the value of \( x \) where \( f(x) \) (the height) is greatest.

If you share the specific quadratic equation or vertex coordinates, I can assist in finding the exact time it takes for the volleyball to reach its maximum height.