Final answer:
The vertex of the quadratic function is (1, 64), representing the maximum height of the ball. The y-intercept is (0, 48), representing the initial height of the ball. The zeros of the function are t = -5 seconds and t = 4 seconds, but only t = 4 seconds is appropriate as it represents the time when the ball hits the ground.
Step-by-step explanation:
The quadratic function h(t) = -16t² + 32t + 48 represents the height of the ball above the ground in feet as a function of time t in seconds. To find the vertex, we need to determine the time at which the ball reaches the maximum height. The vertex of a quadratic function is given by the formula t = -b/2a, where a, b, and c are the coefficients of the quadratic equation. In this case, a = -16 and b = 32, so the vertex time is t = -32/(2*(-16)) = 1 second. Substituting this time into the function, we can find the height at the vertex:
h(1) = -16(1)² + 32(1) + 48 = 64 feet
Therefore, the vertex of the function is (1, 64), which represents the maximum height of the ball.
The y-intercept of the function represents the initial height of the ball at t = 0. To find the y-intercept, we substitute t = 0 into the function:
h(0) = -16(0)² + 32(0) + 48 = 48 feet
So the y-intercept is (0, 48), which represents the initial height of the ball when it is thrown upward from the apartment balcony.
The zeros of the function represent the times at which the ball hits the ground. To find the zeros, we set h(t) = 0 and solve for t:
-16t² + 32t + 48 = 0
Using the quadratic formula, t = (-32 ± √(32² - 4(-16)(48)))/(2(-16))
Simplifying this equation, we get:
t = (-32 ± √(1024 - (-3072)))/-32
t = (-32 ± √4096)/-32
t = (-32 ± 64)/-32
Therefore, the zeros of the function are t = -5 seconds and t = 4 seconds. The positive zero, t = 4 seconds, represents the time at which the ball hits the ground. The negative zero, t = -5 seconds, does not have a physical interpretation in this context as it represents the time when the ball would have passed the top of the building if it had been thrown upward from the ground. Thus, only the zero t = 4 seconds is appropriate in this problem.