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various quadratic equation related problems in IP unit 2

1) Using the quadratic equation x2 - 6x + 8 = 0, perform the following tasks:
a) Solve by factoring.
b) Solve by using the quadratic formula.

2) For the function y = x2 - 6x + 8, perform the following tasks:
a) Put the function in the form y = a(x - h)2 + k.
b) What is the equation for the line of symmetry for the graph of this function?
c) Graph the function using the equation in part a. Explain why it is not necessary to plot points to graph when using y = a (x - h) 2 + k. Show a graph and explanation.
d) In your own words, describe how this graph compares to the graph of y = x2?

3) Suppose a baseball is shot up from the ground straight up with an initial velocity of 32 feet per second. A function can be created by expressing distance above the ground, s, as a function of time, t. This function is s = -16t2 + v0t + s0
? 16 represents ½g, the gravitational pull due to gravity (measured in feet per second 2).
? v0 is the initial velocity (how hard do you throw the object, measured in feet per second).
? s0 is the initial distance above ground (in feet). If you are standing on the ground, then s0 = 0.
a) What is the function that describes this problem?
b) The ball will be how high above the ground after 1 second? and show work
c) How long will it take to hit the ground?
d) What is the maximum height of the ball?

4) Amanda has 400 feet of lumber to frame a rectangular patio (the perimeter of a rectangle is 2 times length plus 2 times width). She wants to maximize the area of her patio (area of a rectangle is length times width). What should the dimensions of the patio be, and show how the maximum area of the patio is calculated from the algebraic equation. Use the vertex form to find the maximum area.

Solution Preview

Please see attached file for detailed solution and graphs.

2. d) For the parabola, we first find its vertex at (3, -1) from (a). Then the graph is symmetric with the line x=3. And from (1) we ...

Solution Summary

The solution is comprised of detailed explanations of various quadratic equation related questions in IP unit 2. It shows how to solve the quadratic equation by factoring and quadratic formula. It also elaborates on how to convert the equation into vertex form and graph the parabola. Moreover, it contains a real life physics example of quadratic equation for a baseball shot up upward. Finally, it explains how to maximize the area of a rectangular patio when the perimeter is fixed using the vertex form in detail.