Question: There are many quadratic equations that you could use to model the distance x and the height y, but we want to find one that is close to reality. Because of this real-life interaction, not every quadratic equation will be acceptable. Let’s consider some of the properties that an appropriate quadratic equation will have. Answer the following questions:

1. The following questions deal with the initial height of the ball:
   1. With respect to the real life situation, what is happening when x=0?
   2. What are the possible values for y when x = 0?
   3. With respect to the graph of the quadratic equation, what is this point called?
2. The following questions deal with the maximum height of the ball:
   1. What is the name of the point on the graph (a parabola) where the maximum height is attained?
   2. Are some values of y unreasonable?
   3. How is this information important for choosing a window for the graph?
3. The following questions deal with the ending values for the flight of the ball:
   1. What would be the height of the ball when x = 452 feet?
   2. This point on the graph has a name. What is it called?
   3. With respect to the quadratic polynomial, this value of x has a name. What is it called?
4. Using the information collected above, explain why the following equations are poor models of the situation. Give coordinates of points on the graph that support your claim.
   1. y = −0.0003x(x − 452).
   2. y = −0.5x ² + 226x + 3.
   3. y = −0.0003x ² + 0.1188x + 3.834.
   4. y = −0.0003x ² + 0.1446x − 4.068.
5. Why is the constant a in the equation y = a x ² + b x + c negative in a reasonable model?

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