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Quadratic Equations and Maximum and Minimum Values

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Note that x^y computes any number to any power (integer, fraction, decimal).

1)Using the quadratic equation x2 - 3x + 2 = 0, perform the following tasks:

a)Solve by factoring.

b)Solve by completing the square.

c)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 line of symmetry?

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 graph here.

Explanation of graphing.

d)In your own words, describe how this graph compares the graph of y = x2?

3)Suppose you throw a baseball straight up at a 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).

·is the initial velocity (how hard do you throw the object, measured in feet per second).

s 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?

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.

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Solution Summary

Quadratic Equations and Maximum and Minimum Values are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

Solution provided by:
Changping Wang, MA
Education
  • BSc , Wuhan Univ. China
  • MA, Shandong Univ.
Recent Feedback
  • "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
  • "excellent work"
  • "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
  • "Thank you"
  • "Thank you very much for your valuable time and assistance!"
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