a ball is thrown upward at 64 feet per second from the top of an 80 feet high building. The height of the ball can be modeled by S(t) = -16t^2 + 64t + 80(feet), where t is the number of seconds after the ball is thrown. describe the graph model

graph of what? Position vs time?

It will be a parabola, starting on one side at t=0 v=64

The velocity slows to zero, then goes negative until the end.

you throw a tennis ball upward from a height of 4 feet with an initial vertical velocity of 36 feet per second.

a. write an equation that models the height h (in feet) of the tennis ball as a function of the time t (in seconds) after its thrown.
b. does the ball reach a height of 30 feet? explain.

The graph model of the height of the ball can be described by the equation S(t) = -16t^2 + 64t + 80, where S(t) represents the height of the ball in feet and t represents the time in seconds.

The equation is a quadratic function in the form of S(t) = at^2 + bt + c, where a, b, and c are constants. In this case, a = -16, b = 64, and c = 80.

The graph of this equation will be a downward-opening parabola since the coefficient of t^2 (a) is negative.

At t = 0, the ball is thrown upwards, so the initial height of the ball is 80 feet, which is represented by the y-intercept of the graph.

As time progresses, the height of the ball decreases until the ball reaches its maximum height. The maximum height can be found by evaluating the vertex of the parabola, which is given by the formula t = -b/2a. In this case, t = -64/(2*-16) = 2 seconds. Substituting t = 2 into the equation, we can find the maximum height reached by the ball.

After reaching the maximum height, the ball starts descending. Eventually, it will hit the ground when its height becomes zero. To find this time, we can set the equation S(t) = 0 and solve for t.

The graph of the model will show the height of the ball as a function of time, and it will be a smooth curve that starts at 80 feet, reaches a maximum height, and then comes back down to zero.

To describe the graph model, let's break it down:

1. S(t): This is the height of the ball at a given time t. It represents the vertical position of the ball above the ground.

2. -16t^2: This term represents the effect of gravity on the ball's height. Since gravity pulls the ball downward, the quadratic term (-16t^2) shows that the height decreases over time at an accelerating rate.

3. 64t: This term represents the velocity of the ball. The coefficient of t (64) indicates that the ball is thrown upward with an initial velocity of 64 feet per second.

4. 80: This constant term represents the initial height of the ball. It indicates that the ball is thrown from the top of an 80-feet high building.

Now, let's consider the graph of this model:

- As the ball is thrown upward, its height initially increases because of the positive coefficient of t^2 (before gravity takes effect).

- The maximum height the ball will reach can be found by finding the vertex of the parabolic curve. The vertex occurs at t = -b/2a, where a and b are the coefficients of the quadratic equation (-16t^2 + 64t + 80). In this case, t = -64/(2*(-16)) = 2 seconds. So, the ball reaches its maximum height after 2 seconds.

- After reaching its peak, the ball starts descending due to gravity. The negative coefficient of t^2 causes the height to decrease at an increasing rate.

- Eventually, the ball will reach the ground (height = 0) when S(t) = 0. This can be found by setting the equation -16t^2 + 64t + 80 = 0 and solving for t.

Overall, the graph of the model will be a downward-opening parabolic curve, starting from the initial height of 80 feet, peaking at some height after 2 seconds, and ultimately reaching a height of 0 when it hits the ground.