Question

your turn (show all your work)
jason jumped off of a cliff into the ocean in acapulco while vacationing with some friends. his height as a function of time could be modeled by the function below:
$f(x)=-16x^{2}+16x + 480$
where \x\ is the time in seconds and f(x) is the height in feet
value of a 16
value of b 16
value of c 480
what is the y - intercept and what does it mean in terms of jasons situation?
find the vertex:
draw a sketch of the situation:
explain what your solution to the vertex means for jason:

Explanation:

Step1: Find the y - intercept

For a quadratic function \(y = ax^{2}+bx + c\), the y - intercept is found by setting \(x = 0\).
Substitute \(x = 0\) into \(f(x)=-16x^{2}+16x + 480\):
\(f(0)=-16(0)^{2}+16(0)+480\)
\(f(0)=480\)
In terms of Jason's situation, when \(x = 0\) (at the moment he jumps), his height is \(480\) feet. So the y - intercept is \((0,480)\) and it means Jason's initial height (height at time \(x = 0\) seconds) is \(480\) feet.

Step2: Find the vertex

The x - coordinate of the vertex of a quadratic function \(y = ax^{2}+bx + c\) is given by \(x=-\frac{b}{2a}\).
Here \(a=-16\) and \(b = 16\).
\(x=-\frac{16}{2\times(-16)}=\frac{-16}{-32}=0.5\)
To find the y - coordinate of the vertex, substitute \(x = 0.5\) into the function:
\(f(0.5)=-16(0.5)^{2}+16(0.5)+480\)
\(=-16\times0.25 + 8+480\)
\(=-4 + 8+480\)
\(=484\)
So the vertex is \((0.5,484)\)

Step3: Explain the vertex

The x - coordinate of the vertex \(x = 0.5\) represents the time (in seconds) when Jason reaches his maximum height. The y - coordinate \(y = 484\) represents the maximum height (in feet) that Jason reaches.

Answer:

• Y - intercept: \((0,480)\). It means Jason's initial height (at the moment he jumps, \(x = 0\) seconds) is \(480\) feet.
• Vertex: \((0.5,484)\)
• Explanation of vertex: At \(x = 0.5\) seconds, Jason reaches his maximum height of \(484\) feet.