The Quadratic Function Model

Oct 13, 2025 by miodeya.com 112 views

**A Company Produces Remote-Controlled Helicopters: A Quadratic Function Model** ===========================================================

Introduction

In this article, we will explore a company that produces remote-controlled helicopters and model their profit as a function of the number of helicopters produced per week using a quadratic function. We will also provide a step-by-step guide on how to create this model and answer some frequently asked questions related to quadratic functions.

The Quadratic Function Model

The company's profit, in thousands of dollars, as a function of the number of helicopters produced per week can be modeled by a quadratic function. When 1 helicopter is produced per week, the company's profit is $10,000. When 2 helicopters are produced per week, the company's profit is $15,000. When 3 helicopters are produced per week, the company's profit is $18,000.

Let's assume that the quadratic function model is in the form of:

f(x) = ax^2 + bx + c

where x is the number of helicopters produced per week and f(x) is the company's profit in thousands of dollars.

Finding the Values of a, b, and c

To find the values of a, b, and c, we can use the given information and create a system of equations.

When x = 1, f(x) = 10 When x = 2, f(x) = 15 When x = 3, f(x) = 18

Substituting these values into the quadratic function model, we get:

a(1)^2 + b(1) + c = 10 a(2)^2 + b(2) + c = 15 a(3)^2 + b(3) + c = 18

Simplifying the equations, we get:

a + b + c = 10 4a + 2b + c = 15 9a + 3b + c = 18

Solving the System of Equations

To solve the system of equations, we can use the method of substitution or elimination. Let's use the elimination method.

Subtracting the first equation from the second equation, we get:

3a + b = 5

Subtracting the first equation from the third equation, we get:

8a + 2b = 8

Multiplying the first equation by 2, we get:

2a + 2b = 20

Subtracting the second equation from the third equation, we get:

6a = -12

Dividing both sides by 6, we get:

a = -2

Substituting a = -2 into the first equation, we get:

-2 + b + c = 10

Simplifying the equation, we get:

b + c = 12

Substituting a = -2 into the second equation, we get:

-8 + 2b + c = 15

Simplifying the equation, we get:

2b + c = 23

Finding the Values of b and c

To find the values of b and c, we can use the two equations:

b + c = 12 2b + c = 23

Subtracting the first equation from the second equation, we get:

b = 11

Substituting b = 11 into the first equation, we get:

11 + c = 12

Simplifying the equation, we get:

c = 1

The Quadratic Function Model

Now that we have found the values of a, b, and c, we can write the quadratic function model as:

f(x) = -2x^2 + 11x + 1

Graphing the Quadratic Function

To graph the quadratic function, we can use a graphing calculator or a computer algebra system. The graph of the quadratic function is a parabola that opens downward.

Interpreting the Graph

The graph of the quadratic function shows that the company's profit increases as the number of helicopters produced per week increases, but at a decreasing rate. The maximum profit occurs when the number of helicopters produced per week is 5.5.

Conclusion

In this article, we have modeled the company's profit as a function of the number of helicopters produced per week using a quadratic function. We have found the values of a, b, and c and written the quadratic function model as f(x) = -2x^2 + 11x + 1. We have also graphed the quadratic function and interpreted the graph.

Frequently Asked Questions

Q: What is a quadratic function?

A: A quadratic function is a polynomial function of degree two, which means that the highest power of the variable is two.

Q: What is the general form of a quadratic function?

A: The general form of a quadratic function is f(x) = ax^2 + bx + c, where a, b, and c are constants.

Q: How do I find the values of a, b, and c in a quadratic function?

A: To find the values of a, b, and c, you can use the given information and create a system of equations.

Q: How do I graph a quadratic function?

A: To graph a quadratic function, you can use a graphing calculator or a computer algebra system.

Q: What does the graph of a quadratic function represent?

A: The graph of a quadratic function represents the relationship between the input and output of the function.

Q: What is the maximum profit of the company?

A: The maximum profit of the company occurs when the number of helicopters produced per week is 5.5.

Q: How do I find the maximum profit of the company?

A: To find the maximum profit of the company, you can use the graph of the quadratic function and find the x-coordinate of the vertex.

Q: What is the x-coordinate of the vertex of a quadratic function?

A: The x-coordinate of the vertex of a quadratic function is given by the formula x = -b/2a.

Q: How do I find the y-coordinate of the vertex of a quadratic function?

A: To find the y-coordinate of the vertex of a quadratic function, you can substitute the x-coordinate into the function.

Q: What does the y-coordinate of the vertex of a quadratic function represent?

A: The y-coordinate of the vertex of a quadratic function represents the maximum or minimum value of the function.

Q: How do I determine whether the vertex of a quadratic function is a maximum or minimum?

A: To determine whether the vertex of a quadratic function is a maximum or minimum, you can look at the sign of the coefficient of the squared term. If the coefficient is positive, the vertex is a minimum. If the coefficient is negative, the vertex is a maximum.