Which Function Can Be Used To Determine The Daily Revenue If The Price Of A Coupon Card Is Decreased By X X X Dollars?A. R ( X ) = − X 2 + 6 X + 40 R(x) = -x^2 + 6x + 40 R ( X ) = − X 2 + 6 X + 40 B. R ( X ) = − ( X − 4 ) ( X + 10 R(x) = -(x - 4)(x + 10 R ( X ) = − ( X − 4 ) ( X + 10 ]C. R(x) = -\left(x^2 - 14x + 40\right ]D.

Introduction

In the world of business and economics, understanding the relationship between price and revenue is crucial for making informed decisions. When the price of a product, such as a coupon card, is decreased, it can lead to an increase in sales, resulting in higher revenue. However, the relationship between price and revenue is not always linear, and it can be affected by various factors. In this article, we will explore the function that can be used to determine the daily revenue if the price of a coupon card is decreased by $x$ dollars.

Understanding the Problem

Let's assume that the original price of the coupon card is $p$ dollars, and the daily revenue is $R$ dollars. When the price is decreased by $x$ dollars, the new price becomes $p - x$ dollars. We want to find the function that represents the daily revenue, $R(x)$, in terms of the decrease in price, $x$.

Analyzing the Options

We are given four options for the function $R(x)$:

A. $R(x) = -x^2 + 6x + 40$ B. $R(x) = -(x - 4)(x + 10)$ C. $R(x) = -\left(x^2 - 14x + 40\right)$ D. (no option provided)

To determine which function is correct, we need to analyze each option and see if it satisfies the conditions of the problem.

Option A: $R(x) = -x^2 + 6x + 40$

Let's start by analyzing option A. This function is a quadratic function, which means it has a parabolic shape. The coefficient of the $x^2$ term is negative, indicating that the function opens downward. This means that as the price decreases, the revenue will increase, which is consistent with our expectations.

To verify if this function is correct, we need to check if it satisfies the conditions of the problem. Let's assume that the original price of the coupon card is $p$ dollars, and the daily revenue is $R$ dollars. When the price is decreased by $x$ dollars, the new price becomes $p - x$ dollars. We want to find the function that represents the daily revenue, $R(x)$, in terms of the decrease in price, $x$.

Substituting $p - x$ for $p$ in the function $R(x) = -x^2 + 6x + 40$, we get:

$R(x) = -(p - x)^2 + 6(p - x) + 40$

Expanding the squared term, we get:

$R(x) = -p^2 + 2px - x^2 + 6p - 6x + 40$

Combining like terms, we get:

$R(x) = -p^2 + (2p - 6)x + 6p + 40$

This function is consistent with our expectations, as it shows that the revenue increases as the price decreases.

Option B: $R(x) = -(x - 4)(x + 10)$

Let's analyze option B. This function is also a quadratic function, but it has a different form. The coefficient of the $x^2$ term is negative, indicating that the function opens downward. This means that as the price decreases, the revenue will increase, which is consistent with our expectations.

To verify if this function is correct, we need to check if it satisfies the conditions of the problem. Substituting $p - x$ for $p$ in the function $R(x) = -(x - 4)(x + 10)$, we get:

$R(x) = -(p - x - 4)(p - x + 10)$

Expanding the product, we get:

$R(x) = -(p^2 - 2px + x^2 - 4p + 10p - 10x - 40)$

Simplifying the expression, we get:

$R(x) = -(p^2 - 2px + x^2 + 6p - 10x - 40)$

This function is not consistent with our expectations, as it shows that the revenue decreases as the price decreases.

Option C: $R(x) = -\left(x^2 - 14x + 40\right)$

Let's analyze option C. This function is also a quadratic function, but it has a different form. The coefficient of the $x^2$ term is negative, indicating that the function opens downward. This means that as the price decreases, the revenue will increase, which is consistent with our expectations.

To verify if this function is correct, we need to check if it satisfies the conditions of the problem. Substituting $p - x$ for $p$ in the function $R(x) = -\left(x^2 - 14x + 40\right)$, we get:

$R(x) = -\left((p - x)^2 - 14(p - x) + 40\right)$

Expanding the squared term, we get:

$R(x) = -\left(p^2 - 2px + x^2 - 14p + 14x + 40\right)$

Simplifying the expression, we get:

$R(x) = -p^2 + 2px - x^2 + 14p - 14x - 40$

This function is not consistent with our expectations, as it shows that the revenue decreases as the price decreases.

Conclusion

In conclusion, the correct function that can be used to determine the daily revenue if the price of a coupon card is decreased by $x$ dollars is option A: $R(x) = -x^2 + 6x + 40$. This function is consistent with our expectations, as it shows that the revenue increases as the price decreases.

Final Answer

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Introduction

In our previous article, we explored the function that can be used to determine the daily revenue if the price of a coupon card is decreased by $x$ dollars. We analyzed four options and determined that option A: $R(x) = -x^2 + 6x + 40$ is the correct function. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the original price of the coupon card?

A: The original price of the coupon card is not specified in the problem. However, we can assume that it is a constant value, denoted by $p$.

Q: How does the function $R(x)$ change when the price of the coupon card is decreased by $x$ dollars?

A: When the price of the coupon card is decreased by $x$ dollars, the new price becomes $p - x$ dollars. The function $R(x)$ represents the daily revenue in terms of the decrease in price, $x$. As the price decreases, the revenue increases, which is consistent with our expectations.

Q: What is the coefficient of the $x^2$ term in the function $R(x)$?

A: The coefficient of the $x^2$ term in the function $R(x)$ is $-1$. This means that the function opens downward, indicating that the revenue increases as the price decreases.

Q: How does the function $R(x)$ change when the price of the coupon card is increased by $x$ dollars?

A: When the price of the coupon card is increased by $x$ dollars, the new price becomes $p + x$ dollars. The function $R(x)$ represents the daily revenue in terms of the increase in price, $x$. As the price increases, the revenue decreases, which is consistent with our expectations.

Q: What is the relationship between the price of the coupon card and the daily revenue?

A: The relationship between the price of the coupon card and the daily revenue is represented by the function $R(x) = -x^2 + 6x + 40$. As the price decreases, the revenue increases, and as the price increases, the revenue decreases.

Q: Can the function $R(x)$ be used to determine the daily revenue if the price of the coupon card is decreased by a percentage?

A: Yes, the function $R(x)$ can be used to determine the daily revenue if the price of the coupon card is decreased by a percentage. To do this, we need to convert the percentage decrease to a dollar amount, and then use the function $R(x)$ to determine the daily revenue.

Q: Can the function $R(x)$ be used to determine the daily revenue if the price of the coupon card is increased by a percentage?

A: Yes, the function $R(x)$ can be used to determine the daily revenue if the price of the coupon card is increased by a percentage. To do this, we need to convert the percentage increase to a dollar amount, and then use the function $R(x)$ to determine the daily revenue.

Conclusion

In conclusion, the function $R(x) = -x^2 + 6x + 40$ can be used to determine the daily revenue if the price of a coupon card is decreased by $x$ dollars. We answered some frequently asked questions related to this topic and provided additional information to help clarify the relationship between the price of the coupon card and the daily revenue.

Final Answer

The final answer is option A: $R(x) = -x^2 + 6x + 40$.