The Quadratic Function Y = − 10 X 2 + 160 X − 430 Y = -10x^2 + 160x - 430 Y = − 10 X 2 + 160 X − 430 Models A Store's Daily Profit ( Y Y Y ), In Dollars, For Selling T-shirts Priced At X X X Dollars.Match Each Item With What It Represents In This Situation By Entering The

Introduction

In the world of mathematics, quadratic functions are used to model various real-world situations, including the profit of a store selling T-shirts. The quadratic function $y = -10x^2 + 160x - 430$ represents the daily profit of a store in dollars, for selling T-shirts priced at $x$ dollars. In this article, we will match each item with what it represents in this situation.

Understanding the Quadratic Function

The quadratic function $y = -10x^2 + 160x - 430$ is a polynomial of degree two, which means it has a highest power of two. The general form of a quadratic function is $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. In this case, $a = -10$, $b = 160$, and $c = -430$.

Coefficients of the Quadratic Function

• $a = -10$: The coefficient of the squared term, $x^2$, represents the rate of change of the profit with respect to the price of the T-shirt. A negative value indicates that the profit decreases as the price increases.
• $b = 160$: The coefficient of the linear term, $x$, represents the rate of change of the profit with respect to the price of the T-shirt. A positive value indicates that the profit increases as the price increases.
• $c = -430$: The constant term represents the initial profit of the store when the price of the T-shirt is zero.

Graphing the Quadratic Function

To visualize the relationship between the price of the T-shirt and the daily profit, we can graph the quadratic function. The graph of a quadratic function is a parabola, which is a U-shaped curve.

Graphing the Quadratic Function

• Vertex: The vertex of the parabola represents the maximum or minimum value of the function. In this case, the vertex is the point where the profit is maximum.
• Axis of Symmetry: The axis of symmetry is the vertical line that passes through the vertex and divides the parabola into two equal parts.

Matching Items with What They Represent

Now that we have a good understanding of the quadratic function and its graph, let's match each item with what it represents in this situation.

Matching Items

• Price of the T-shirt ($x$): The price of the T-shirt is represented by the variable $x$.
• Daily Profit ($y$): The daily profit of the store is represented by the variable $y$.
• Rate of Change of Profit with Respect to Price ($a$): The rate of change of the profit with respect to the price of the T-shirt is represented by the coefficient $a = -10$.
• Rate of Change of Profit with Respect to Price ($b$): The rate of change of the profit with respect to the price of the T-shirt is represented by the coefficient $b = 160$.
• Initial Profit ($c$): The initial profit of the store is represented by the constant term $c = -430$.

Conclusion

In conclusion, the quadratic function $y = -10x^2 + 160x - 430$ models a store's daily profit for selling T-shirts priced at $x$ dollars. By understanding the coefficients of the quadratic function and its graph, we can match each item with what it represents in this situation.

Key Takeaways

• The quadratic function $y = -10x^2 + 160x - 430$ represents the daily profit of a store in dollars, for selling T-shirts priced at $x$ dollars.
• The coefficient $a = -10$ represents the rate of change of the profit with respect to the price of the T-shirt.
• The coefficient $b = 160$ represents the rate of change of the profit with respect to the price of the T-shirt.
• The constant term $c = -430$ represents the initial profit of the store.

Real-World Applications

The quadratic function $y = -10x^2 + 160x - 430$ has various real-world applications, including:

• Optimization: The quadratic function can be used to optimize the price of the T-shirt to maximize the daily profit.
• Prediction: The quadratic function can be used to predict the daily profit of the store based on the price of the T-shirt.

Future Research Directions

Future research directions include:

• Exploring Other Quadratic Functions: Exploring other quadratic functions that model real-world situations, such as the profit of a store selling other products.
• Developing Optimization Algorithms: Developing optimization algorithms to optimize the price of the T-shirt to maximize the daily profit.

References

• [1] "Quadratic Functions" by Math Open Reference. Retrieved from https://www.mathopenref.com/quadratic.html
• [2] "Graphing Quadratic Functions" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra2/quadratic-functions-and-graphs

Appendix

The quadratic function $y = -10x^2 + 160x - 430$ can be graphed using various graphing tools, including:

• Graphing Calculator: A graphing calculator can be used to graph the quadratic function and visualize the relationship between the price of the T-shirt and the daily profit.
• Computer Algebra System: A computer algebra system, such as Mathematica or Maple, can be used to graph the quadratic function and visualize the relationship between the price of the T-shirt and the daily profit.
  Quadratic Function Modeling a Store's Daily Profit: Q&A ===========================================================

Introduction

In our previous article, we explored the quadratic function $y = -10x^2 + 160x - 430$ that models a store's daily profit for selling T-shirts priced at $x$ dollars. In this article, we will answer some frequently asked questions (FAQs) related to this topic.

Q&A

Q: What is the purpose of the quadratic function in this situation?

A: The quadratic function $y = -10x^2 + 160x - 430$ is used to model the daily profit of a store for selling T-shirts priced at $x$ dollars. It helps to visualize the relationship between the price of the T-shirt and the daily profit.

Q: What does the coefficient $a = -10$ represent?

A: The coefficient $a = -10$ represents the rate of change of the profit with respect to the price of the T-shirt. A negative value indicates that the profit decreases as the price increases.

Q: What does the coefficient $b = 160$ represent?

A: The coefficient $b = 160$ represents the rate of change of the profit with respect to the price of the T-shirt. A positive value indicates that the profit increases as the price increases.

Q: What does the constant term $c = -430$ represent?

A: The constant term $c = -430$ represents the initial profit of the store when the price of the T-shirt is zero.

Q: How can the quadratic function be used in real-world applications?

A: The quadratic function $y = -10x^2 + 160x - 430$ can be used in various real-world applications, including optimization and prediction. For example, it can be used to optimize the price of the T-shirt to maximize the daily profit or to predict the daily profit of the store based on the price of the T-shirt.

Q: What are some future research directions related to this topic?

A: Some future research directions related to this topic include exploring other quadratic functions that model real-world situations, developing optimization algorithms to optimize the price of the T-shirt to maximize the daily profit, and using machine learning techniques to predict the daily profit of the store based on the price of the T-shirt.

Q: How can the quadratic function be graphed?

A: The quadratic function $y = -10x^2 + 160x - 430$ can be graphed using various graphing tools, including graphing calculators and computer algebra systems.

Q: What are some common mistakes to avoid when working with quadratic functions?

A: Some common mistakes to avoid when working with quadratic functions include:

• Not checking the domain of the function: Make sure to check the domain of the function to ensure that it is defined for all values of $x$.
• Not checking the range of the function: Make sure to check the range of the function to ensure that it is defined for all values of $y$.
• Not using the correct formula for the axis of symmetry: Make sure to use the correct formula for the axis of symmetry, which is $x = -\frac{b}{2a}$.

Conclusion

In conclusion, the quadratic function $y = -10x^2 + 160x - 430$ models a store's daily profit for selling T-shirts priced at $x$ dollars. By understanding the coefficients of the quadratic function and its graph, we can answer some frequently asked questions related to this topic.

Key Takeaways

• The quadratic function $y = -10x^2 + 160x - 430$ represents the daily profit of a store for selling T-shirts priced at $x$ dollars.
• The coefficient $a = -10$ represents the rate of change of the profit with respect to the price of the T-shirt.
• The coefficient $b = 160$ represents the rate of change of the profit with respect to the price of the T-shirt.
• The constant term $c = -430$ represents the initial profit of the store when the price of the T-shirt is zero.

Real-World Applications

The quadratic function $y = -10x^2 + 160x - 430$ has various real-world applications, including:

• Optimization: The quadratic function can be used to optimize the price of the T-shirt to maximize the daily profit.
• Prediction: The quadratic function can be used to predict the daily profit of the store based on the price of the T-shirt.

Future Research Directions

Future research directions include:

• Exploring Other Quadratic Functions: Exploring other quadratic functions that model real-world situations, such as the profit of a store selling other products.
• Developing Optimization Algorithms: Developing optimization algorithms to optimize the price of the T-shirt to maximize the daily profit.
• Using Machine Learning Techniques: Using machine learning techniques to predict the daily profit of the store based on the price of the T-shirt.

References

• [1] "Quadratic Functions" by Math Open Reference. Retrieved from https://www.mathopenref.com/quadratic.html
• [2] "Graphing Quadratic Functions" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra2/quadratic-functions-and-graphs

Appendix

The quadratic function $y = -10x^2 + 160x - 430$ can be graphed using various graphing tools, including:

• Graphing Calculator: A graphing calculator can be used to graph the quadratic function and visualize the relationship between the price of the T-shirt and the daily profit.
• Computer Algebra System: A computer algebra system, such as Mathematica or Maple, can be used to graph the quadratic function and visualize the relationship between the price of the T-shirt and the daily profit.