Question: There are different algebraic ways to find a reasonable quadratic model of the situation. We have some information about the path of the ball, giving some information about points of its graph (a parabola).

Suppose the ball was 2 feet above ground when Ernie Thayer hit it, and that it reached a maximum height of approximately 22.74 feet when it was approximately 220.8 feet away from where he hit the ball. The ball lands after traveling a ground distance of approximately 452 feet.

We will find equations to model the situation by using two algebraic methods. (Show all your work for each part.)

1. Find an equation of the form y = C(x − z ₁ ) (x − z ₂ ) where z ₁ and z ₂ are the zeros (or roots) of the quadratic polynomial (or x-intercepts of the graph) and C is a scaling constant that needs to be determined.

1. Find the other root. (Hint: Use the known root, the vertex, and a symmetry property of the graph.)
2. Find the constant C. (Round this value to two significant positions. [Leading zeros are not significant, but trailing zeros are significant.] For example, 0.000347 would be rounded to 0.00035 and 0.000301 would be rounded to 0.00030. Hint: To find C, you can use the initial height.)
   iii. Write out the equation that you found and algebraically check that it satisfies all
   the necessary conditions. (Note: Show your work! Expect small errors because of rounding.)
   (b) Find an equation of the form y = A (x − h) ² + k where the vertex is at (h, k) and the constant A is a scaling factor.
   1. Based on the information that you were given, what are the coordinates of the vertex.
   2. Find the constant A. (Round this value to two significant positions.)
   3. Write out the equation that you found and algebraically check that it satisfies all the necessary conditions. (Note: Show your work! Expect small errors because of rounding.)
      
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