Understanding the Components of the Profit Function

Oct 13, 2025 by leolavanderia.com.br 44 views

Introduction

In the world of business, understanding the profit function is crucial for making informed decisions. A home-based sign company, for instance, needs to determine the optimal price for its signs to maximize profits. In this article, we will delve into the profit function of a home-based sign company and explore how to calculate its profit.

The Profit Function

The profit function of the home-based sign company is given by the quadratic equation:

p(x) = -10x^2 + 498x - 1,500

where $x$ is the price of each sign sold.

Understanding the Components of the Profit Function

To understand the profit function, let's break down its components:

Quadratic term: The quadratic term, $-10x^2$ , represents the decrease in profit as the price of each sign increases. The coefficient, $-10$ , indicates that the profit decreases at a rate of $10$ units for every $1$ unit increase in price.
Linear term: The linear term, $498x$ , represents the increase in profit as the price of each sign increases. The coefficient, $498$ , indicates that the profit increases at a rate of $498$ units for every $1$ unit increase in price.
Constant term: The constant term, $-1,500$ , represents the initial profit of the company.

Calculating the Company's Profit

To calculate the company's profit, we need to substitute the price of each sign into the profit function. Let's assume the company sells each sign at a price of $x$ dollars.

Step 1: Substitute the price into the profit function

p(x) = -10x^2 + 498x - 1,500

Step 2: Simplify the expression

p(x) = -10x^2 + 498x - 1,500

Step 3: Evaluate the expression

To evaluate the expression, we need to substitute a specific value for $x$ . Let's assume the company sells each sign at a price of $x = 50$ dollars.

p(50) = -10(50)^2 + 498(50) - 1,500

Step 4: Calculate the profit

p(50) = -10(2,500) + 24,900 - 1,500

p(50) = -25,000 + 24,900 - 1,500

p(50) = -1,600

Conclusion

In conclusion, the profit function of the home-based sign company is given by the quadratic equation:

p(x) = -10x^2 + 498x - 1,500

To calculate the company's profit, we need to substitute the price of each sign into the profit function and simplify the expression. In this example, we assumed the company sells each sign at a price of $x = 50$ dollars and calculated the profit to be $-1,600$ dollars.

Optimizing the Price of Each Sign

To maximize profits, the company needs to determine the optimal price for each sign. This can be done by finding the vertex of the profit function, which represents the maximum profit.

Finding the Vertex of the Profit Function

To find the vertex of the profit function, we need to complete the square:

p(x) = -10x^2 + 498x - 1,500

p(x) = -10(x^2 - 49.8x) - 1,500

p(x) = -10(x^2 - 49.8x + 249.6) - 1,500 + 2,496

p(x) = -10(x - 24.9)^2 + 996

Vertex Form

The vertex form of the profit function is:

p(x) = -10(x - 24.9)^2 + 996

Vertex

The vertex of the profit function is:

(24.9, 996)

Conclusion

In conclusion, the vertex of the profit function represents the maximum profit, which occurs when the price of each sign is $24.9$ dollars. To maximize profits, the company should sell each sign at a price of $24.9$ dollars.

Graphing the Profit Function

To visualize the profit function, we can graph it using a graphing calculator or a computer algebra system.

Graph

The graph of the profit function is a parabola that opens downward, indicating that the profit decreases as the price of each sign increases.

X-Intercepts

The x-intercepts of the profit function represent the prices at which the company breaks even.

Finding the X-Intercepts

To find the x-intercepts, we need to set the profit function equal to zero and solve for $x$ :

-10x^2 + 498x - 1,500 = 0

Solving the Quadratic Equation

To solve the quadratic equation, we can use the quadratic formula:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

where $a = -10$ , $b = 498$ , and $c = -1,500$ .

Solving for x

x = \frac{-498 \pm \sqrt{498^2 - 4(-10)(-1,500)}}{2(-10)}

x = \frac{-498 \pm \sqrt{248,004 - 60,000}}{-20}

x = \frac{-498 \pm \sqrt{188,004}}{-20}

x = \frac{-498 \pm 434.2}{-20}

X-Intercepts

The x-intercepts of the profit function are:

x = \frac{-498 + 434.2}{-20} = 6.36

x = \frac{-498 - 434.2}{-20} = 16.61

Conclusion

In conclusion, the x-intercepts of the profit function represent the prices at which the company breaks even. To break even, the company should sell each sign at a price of $6.36$ dollars or $16.61$ dollars.

Sensitivity Analysis

To determine the sensitivity of the profit function to changes in the price of each sign, we can perform a sensitivity analysis.

Sensitivity Analysis

To perform a sensitivity analysis, we need to calculate the partial derivative of the profit function with respect to the price of each sign:

\frac{\partial p}{\partial x} = -20x + 498

Evaluating the Partial Derivative

To evaluate the partial derivative, we need to substitute a specific value for $x$ . Let's assume the company sells each sign at a price of $x = 50$ dollars.

\frac{\partial p}{\partial x} = -20(50) + 498

\frac{\partial p}{\partial x} = -1,000 + 498

\frac{\partial p}{\partial x} = -502

Conclusion

In conclusion, the partial derivative of the profit function with respect to the price of each sign is $-502$ . This indicates that the profit decreases at a rate of $502$ units for every $1$ unit increase in price.

Conclusion

In conclusion, the profit function of the home-based sign company is given by the quadratic equation:

p(x) = -10x^2 + 498x - 1,500

To maximize profits, the company should sell each sign at a price of $24.9$ dollars. The x-intercepts of the profit function represent the prices at which the company breaks even. To break even, the company should sell each sign at a price of $6.36$ dollars or $16.61$ dollars.

Introduction

In our previous article, we explored the profit function of a home-based sign company and how to calculate its profit. In this article, we will answer some frequently asked questions about the profit function and provide additional insights into the world of business.

Q: What is the profit function, and why is it important?

A: The profit function is a mathematical equation that represents the relationship between the price of a product and the profit earned by a company. It is an essential tool for businesses to determine the optimal price for their products and maximize profits.

Q: How do I calculate the profit function for my business?

A: To calculate the profit function, you need to determine the relationship between the price of your product and the profit earned by your company. This can be done by analyzing your company's financial data and identifying the key factors that affect your profit.

Q: What is the vertex of the profit function, and why is it important?

A: The vertex of the profit function represents the maximum profit that a company can earn. It is an important concept in business because it helps companies determine the optimal price for their products and maximize profits.

Q: How do I find the vertex of the profit function?

A: To find the vertex of the profit function, you need to complete the square and rewrite the equation in vertex form. This will give you the coordinates of the vertex, which represent the maximum profit.

Q: What are the x-intercepts of the profit function, and why are they important?

A: The x-intercepts of the profit function represent the prices at which a company breaks even. They are important because they help companies determine the minimum price at which they can sell their products and still earn a profit.

Q: How do I perform a sensitivity analysis on the profit function?

A: To perform a sensitivity analysis on the profit function, you need to calculate the partial derivative of the profit function with respect to the price of each sign. This will give you the rate at which the profit changes in response to changes in the price.

Q: What is the partial derivative of the profit function, and why is it important?

A: The partial derivative of the profit function represents the rate at which the profit changes in response to changes in the price. It is an important concept in business because it helps companies determine the sensitivity of their profit to changes in the price.

Q: How do I use the profit function to make business decisions?

A: The profit function can be used to make a variety of business decisions, including determining the optimal price for a product, identifying the key factors that affect profit, and performing sensitivity analysis. By using the profit function, businesses can make informed decisions that maximize profits and drive growth.

Conclusion

In conclusion, the profit function is a powerful tool for businesses to determine the optimal price for their products and maximize profits. By understanding the profit function and how to calculate it, businesses can make informed decisions that drive growth and profitability.

Additional Resources

For more information on the profit function and how to use it to make business decisions, check out the following resources:

Profit Function Calculator: A calculator that allows you to input the price of a product and calculate the profit earned by a company.
Profit Function Tutorial: A tutorial that provides step-by-step instructions on how to calculate the profit function and use it to make business decisions.
Profit Function Examples: Examples of how to use the profit function to make business decisions, including determining the optimal price for a product and performing sensitivity analysis.