Hello everyone, I hope you're having a wonderful day. I'm struggling with this problem; I know it's about quadratic functions, but I'm not sure where to begin, and I'm overall confused. I would appreciate it if you could all help me solve this problem!

**Problem:**

Vernon Wells hits a baseball that travels for 142 m before it lands. The flight of the ball can be modeled by a quadratic function in which x is the horizontal distance the ball has traveled away from Vernon, and h(x) is the height of the ball at that distance.

There are many quadratic equations you could use to model the distance and height, but you want to find one that is close to reality.

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**A. Assume that the ball was between 0.6 m and 1.5 m above the ground when it was hit.**

a) What would h(142) be? *My answer here is 0*

b) What happens when x =0

c) What are the possible values for h(x) when x = 0?

d) What would be a good range of values for the height of the ball? Are some values for the height unreasonable?

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**B. Explain why each function is not a good model of the situation, and support your claim with reasons and a well-labelled sketch.**

a) h(x)= -0.5x(x-142)

b) h(x)= -0.5x2 +71x +1

c) h(x)= -0.0015x2 +0.213x +1.2

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**C. Determine an equation that models the path of the ball, given this additional information:**

*I'm not sure how you'd write a quadratic equation with this information.*

a) The ball was 1.2 m off the ground when it was hit.

b) The ball reached a maximum height of 17 m when it was approximately 70 m away from Vernon. Explain the method you are using to get the equation, and show all of your steps. Why did you approach the problem this way?

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**D. Use the model you created to graph the flight of Vernon’s ball.**
I think part A(d) is probably asking for some plausible values for the maximum height of the ball during its flight. There are many possibilities but they are I think hinting that some would be very unrealistic.

Part B is focusing on the idea that a good model needs to have reasonable height value when x = 0 and also when ball is at its maximum height.

Part C would be easiest to answer if you know vertex form of graph of parabola:

y = a(x – h)^2 + k

That has vertex (h, k), which you know the numbers of. But to solve for a you need to plug in a point you know such as (0, 1.2) or (142, 0).