Question

A ball is thrown into the air its height h(x), in feet, after x seconds is given by the function h(x)= -16(x-1)²+23. What does the vertex indicate about the ball?

A. The ball starts at 1 foot above the ground.
B. The ball starts at 23 feet above the ground.
C. After 1 seconds, the ball is 23 feet above the ground.
D. After 23 seconds, the ball is 1 foot above the ground.

Answer 1

Answer: The vertex of the function h(x)=-16(x-1)²+23 is (1, 23).

This means that the ball reaches its maximum height of 23 feet after 1 second, and then starts to fall back down. The vertex also represents the highest point of the ball's trajectory, also known as the maximum value of the function.

Step-by-step explanation: The given function h(x) represents the height of a ball in feet after x seconds, and it is given by:

h(x) = -16(x - 1)² + 23

This is a quadratic function in standard form, where the coefficient of x^2 is negative, which means that the graph of the function is a downward-facing parabola. The vertex of this parabola is given by the formula:

Vertex = (-b/2a, f(-b/2a))

where a = -16, b = 0, and c = 23 are the coefficients of the quadratic function. Substituting these values into the formula, we get:

Vertex = (-b/2a, f(-b/2a))

= (-0/2(-16), f(0/2(-16)))

= (0, 23)

Therefore, the vertex of the parabolic function h(x) is (0, 23), which indicates that the ball reaches its maximum height of 23 feet after 1 second, and then starts to fall back to the ground.