Question

PROJECTILE A firework is launched from the ground. After 4 seconds, it reaches a

maximum height of 256 feet before returning to the ground 8 seconds after it was
launched. The height of the firework f(x), in feet, after x seconds can be modeled
by a quadratic function.
a. What are the zeros and vertex of f(x)?
b. Sketch a graph of f(x) using the zeros and vertex of the function. Interpret the
key features of the function in the context of the situation.
c. Write a quadratic function that represents the situation.

Answer 1

The zeros of the quadratic function are x = 0 and x = 8. The vertex of the function is at x = -4 and the maximum height of the firework is 256 feet. The quadratic function that represents the situation is f(x) = -x² + 8x.

Step-by-step explanation:

a. To find the zeros of the quadratic function, we need to find the points where the height of the firework is equal to 0. Since the firework returns to the ground 8 seconds after it was launched, the zeros occur at x = 0 and x = 8. The vertex of the quadratic function can be found using the formula x = -b/2a, where a, b, and c are the coefficients of the quadratic function. In this case, the vertex occurs at x = -8/2 = -4. The maximum height of the firework occurs at the vertex, which is 256 feet.

b. Using the zeros and vertex of the function, we can sketch a graph of f(x). The graph will be a parabola that opens downwards and intersects the x-axis at x = 0 and x = 8. The vertex will be the highest point of the graph. The key features of the function in the context of the situation are that the firework reaches a maximum height of 256 feet and returns to the ground 8 seconds after it was launched.

c. The quadratic function that represents the situation is f(x) = -x² + 8x. This function models the height of the firework over time, where x is the time in seconds and f(x) is the height in feet.