Real World Quadratic Functions

1. Solve problem 56 on pages 666-667 of Elementary and Intermediate Algebra.

Reference :
Dugopolski, M. (2012). Elementary and intermediate algebra (4th ed.). New York, NY: McGraw-Hill Publishing.

2. Write a two page paper (not including the title page and reference page) that is formatted in APA style and according to the Math Writing Guide. (must include an introduction and conclusion paragraph) Format your math work as shown in the example, and be concise in your reasoning. In the body of your essay, do the following:

o Refer to the graph you sketched for problem 56. Describe the basic shape of the graph and where it is located in the Cartesian plane. Describe what the graph represents.

o Demonstrate your solution for both the number of clerks that will maximize the profit as well as the total maximum profit possible, making sure to include all mathematical work and an explanation for each step.

o Analyze why this information is important for managers to know, and explain what could happen if the ideal conditions were not met.
Problem 56.
Maximum profit. A chain store manager has been told by the main office that daily profit, P, is related to the number of clerks working that day, x, according to the function P = -25×2 + 300x. What number of clerks will maximize the profit, and what is the maximum possible profit?

Running head: QUADRATIC FUNCTIONS 1
Running header should use a shortened version of the title if the title is long. Page number is located at right margin.
(full title; centered horizontally & vertically) Real World Quadratic Functions
John Q. Student
MAT 222 Week 4 Assignment
Instructor’s Name
Date
QUADRATIC FUNCTIONS 2
Real World Quadratic Functions (title required on first line)
Quadratic functions are perhaps the best example of how math concepts can be
combined into a single problem. To solve these, rules for order of operations, solving equations, exponents, and radicals must be used. Because multiple variables are involved and affect the outcome, quadratics are extension of functions as well.
The following example is similar to #56 on page 666 (Dugopolski, 2012) and is in the form ax2 + bx + c = 0. The profit function P(x) = -12×2 + 600x where x is the number of clerks working. The x-intercepts of the parabola can be found by solving -12×2 + 600x = 0.
-12×2 + 600x = 0 Divide both sides by -1.
12×2 – 600x = 0 Factor the left side.
12x(x – 50) = 0 Use Zero Factor Property
12x = 0 or x – 50 = 0 Solve each equation.
x = 0 or x = 50 The parabola will cross the x-axis at 0 and 50.
This quadratic function has a large a value which means the parabola will be narrow. It also has a negative a value so the parabola will open downward. This means there will be a maximum value of the graph at the vertex, which will happen at the x value of –b/(2a), where
b = 600 and a = -12 in this case.
What number of clerks will maximize the profit?
–b/(2a)
-600/2(-12) Notice the b value is now negative because of the negative in the formula.
-600/-24
25 25 clerks working will maximize profit.
What is the maximum possible profit when this many clerks are working?
QUADRATIC FUNCTIONS 3
P(x) = -12×2 + 600x Start with original profit function to find P(25). P(25) = -12(25)2 + 600(25) Substitute 25 for all x’s in the problem.
P(25) = -12(625) + 15000 The exponent must be solved first.
P(25) = -7500 + 15000
P(25) = $7500 They can expect a possible $7500 profit when 25 clerks are working.
Basically, the graph shows there will be no profit made when zero clerks are working or when 50 clerks are working. The maximum profit will occur when 25 clerks are working. The graph of this function is only relevant in the first quadrant because negative clerks cannot exist.
Conclusion paragraph would go here. Remember to include 4-5 sentences to make a complete paragraph.
QUADRATIC FUNCTIONS 4
Reference
Dugopolski, M. (2012). Elementary and intermediate algebra (4th ed.). New York, NY: McGraw-Hill Publishing.
Use the word ‘Reference’ or ‘References’ as the title.
Text should ALWAYS be included in every assignment! Be sure to use appropriate indentation
(hanging), font (Arial or Times New Roman), and size (12).