Question 599183: Ernie Thayer1 hits a baseball and it travels for 409.6 feet before it lands. When he hits the ball, the ball is between about 2 feet and about 5 feet high. If we ignore air resistance, then physics tells us that the flight of the baseball can be modelled using a quadratic equation of the form:
y=ax2 +bx+c, where x is the horizontal distance that the ball has travelled, and y is the height
of the ball at the given distance. Background: To determine the coefficients in a polynomial equation of degree
n:
y=a0xn +a1xn-1 +a2xn-2 +···+an-2x2 +an-1x+an,
one generally to know n + 1 points. With additional information about the poly- nomial, the number of points that are needed can sometimes be reduced. (This additional information is sometimes called symmetry.)
In addition, if we know certain kinds of points, then there may be special forms of the polynomial equation that are easier to work with. For example, if we know that a cubic has its zeros at 1, 2 and 3, then we can use the three x-intercepts form of the cubic to simplify our work in particular, we know that the cubic has the following form:
y = a(x - 1)(x - 2)(x - 3).
We still need a fourth point on the cubic to determine the value of the constant a.
In this lab, we will focus on 2nd degree or quadratic polynomial equations:
y=ax2 +bx+c.
1. There are many quadratic equations that you could use to model the dis- tance x and the height y, but we want to find one that is realistic. Because of this real-life interaction, not every quadratic equation will be acceptable. Lets consider some of the properties that an appropriate quadratic equation will have. Answer the following questions:
1.
(a) The following questions deal with the initial height of the ball:
With respect to the real life situation, what is happening when x = 0?
What are the possible values for y when x = 0?
With respect to the graph of the quadratic equation, what is this
point called?
(b) The following questions deal with the maximum height of the ball:
What is the name of the point on the graph (a parabola) where the maximum height is attained?
Are some values of y unreasonable?
How is this information important for choosing a window for the
graph?
(c) The following questions deal with the ending values for the flight of the ball:
What would be the height of the ball when x = 409.6 feet?
This point on the graph has a name. What is it called?
With respect to the quadratic polynomial, this value of x has a name. What is it called?
(d) Using the information collected above, explain why the following equa- tions are poor models of the situation. Give coordinates of points on the graph that support your claim.
i. y = -0.002x(x - 409.6). ii. y = -0.5x2 +216x+3.
iii. y = -0.002x2 + 0.879x + 3.981. iv. y = -0.002x2 + 0.8732x - 3.981.
(e) Why is the constant a in the equation y = ax2 +bx+c negative in a reasonable model?
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Part II. Using given information about the graph to find the quadratic equation.
2. 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 4.56 feet above ground when Ernie Thayer hit it, and that it reached a maximum height of approximately 108.42 feet when it was approximately 202.6 feet away from where he hit the ball. The ball lands after travelling a ground distance of approximately 409.6 feet.
We will find equations to model the situation by using two algebraic meth- ods. (Show all your work for each part.)
(a) Find an equation of the form y = C(x - z1)(x - z2) where z1 and z2 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.
Find the other root. (Hint: Use the known root, the vertex, and a symmetry property of the graph.)
Find the constant C. (Round this value to three significant posi- tions. Leading zeros are not significant, but trailing zeros are signif- icant. For example, 0.0003478 would be rounded to 0.000348 and 0.0003501 would be rounded to 0.000350. Hint: To find C, you can use the initial height.)
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.
Based on the information that you were given, what are the coordi- nates of the vertex.
Find the constant A. (Round this value to three significant positions.)
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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Part III. Discussion of when algebra can be used to find a quadratic equation.
3. In part II, we used two different methods to find a quadratic equation to model the situation. Each required different information about the graph.
1.
(a) Can you use algebra to find a quadratic equation if you know the coor- dinates of just one point on its graph?
2.
(b) Can you use algebra to find a quadratic equation if you know the co- ordinates of just two points on its graph? If so, what information is needed?
3.
(c) Rounding errors in the data might result in your finding different cor- rect equations for your answers in parts (a) and (b) above for Problem 2. What characteristic of the data causes this to happen?
Answer by richard1234(7193)   (Show Source):
You can put this solution on YOUR website! You shouldn't post giant homework assignments here. We're here for homework help, not just to do your entire assignment.
Do as much as you can first, then post specific questions on the ones you don't understand.
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