The Final Cost Of A Sale Item Is Determined By Multiplying The Price On The Tag By $75\%$. Which Best Describes The Function That Represents The Situation?$\[ \begin{array}{|c|c|} \hline \text{Price On The Tag, } X & \text{Final Cost}

Introduction

In the world of sales and discounts, understanding the final cost of an item is crucial for both customers and businesses. A common practice is to offer a discount by multiplying the price on the tag by a certain percentage. In this scenario, we are given that the final cost of a sale item is determined by multiplying the price on the tag by $75\%$. This article aims to explore the function that represents this situation and provide a deeper understanding of the mathematical concept involved.

Understanding the Problem

Let's break down the problem statement: "The final cost of a sale item is determined by multiplying the price on the tag by $75\%$". This means that if the price on the tag is $x$, the final cost will be $75\%$ of $x$. Mathematically, this can be represented as:

$$\text{Final Cost} = 0.75x$$

Representing the Function

To represent the function that describes the situation, we need to identify the input and output variables. In this case, the input variable is the price on the tag ($x$), and the output variable is the final cost. The function that represents the situation is a linear function, which can be written in the form:

$$y = mx + b$$

where $y$ is the output variable (final cost), $x$ is the input variable (price on the tag), $m$ is the slope, and $b$ is the y-intercept.

In this case, the slope ($m$) represents the rate of change of the final cost with respect to the price on the tag. Since the final cost is determined by multiplying the price on the tag by $75\%$, the slope is $0.75$. The y-intercept ($b$) represents the value of the final cost when the price on the tag is zero. Since the final cost is always positive, the y-intercept is zero.

Therefore, the function that represents the situation is:

$$y = 0.75x$$

Graphing the Function

To visualize the function, we can graph it on a coordinate plane. The graph of the function will be a straight line with a slope of $0.75$ and a y-intercept of zero.

Interpreting the Graph

The graph of the function represents the relationship between the price on the tag and the final cost. The x-axis represents the price on the tag, and the y-axis represents the final cost. The graph shows that as the price on the tag increases, the final cost also increases at a rate of $75\%$.

Conclusion

In conclusion, the function that represents the situation is a linear function of the form $y = 0.75x$. This function describes the relationship between the price on the tag and the final cost, where the final cost is determined by multiplying the price on the tag by $75\%$. Understanding this function is crucial for both customers and businesses to make informed decisions about sales and discounts.

Key Takeaways

• The final cost of a sale item is determined by multiplying the price on the tag by $75\%$.
• The function that represents the situation is a linear function of the form $y = 0.75x$.
• The graph of the function represents the relationship between the price on the tag and the final cost.
• Understanding this function is crucial for both customers and businesses to make informed decisions about sales and discounts.

Further Exploration

For further exploration, you can try the following:

• Graph the function using different scales and axes.
• Find the equation of the function in terms of the final cost ($y$) as the input variable.
• Use the function to calculate the final cost of different items with varying prices on the tag.
• Explore the concept of discounts and how they affect the final cost of an item.
  The Final Cost of a Sale Item: Q&A =====================================

Introduction

In our previous article, we explored the function that represents the situation where the final cost of a sale item is determined by multiplying the price on the tag by $75\%$. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the formula for calculating the final cost of a sale item?

A: The formula for calculating the final cost of a sale item is:

$$\text{Final Cost} = 0.75x$$

where $x$ is the price on the tag.

Q: How does the final cost change when the price on the tag increases?

A: When the price on the tag increases, the final cost also increases at a rate of $75\%$. This means that if the price on the tag is doubled, the final cost will also be doubled.

Q: What is the y-intercept of the function that represents the situation?

A: The y-intercept of the function that represents the situation is zero. This means that when the price on the tag is zero, the final cost is also zero.

Q: Can the final cost be negative?

A: No, the final cost cannot be negative. Since the final cost is determined by multiplying the price on the tag by $75\%$, it will always be a positive value.

Q: How does the function change if the discount is changed to $50\%$?

A: If the discount is changed to $50\%$, the function that represents the situation will change to:

$$\text{Final Cost} = 0.5x$$

This means that the final cost will be half of the price on the tag.

Q: Can the function be used to calculate the price on the tag from the final cost?

A: Yes, the function can be used to calculate the price on the tag from the final cost. To do this, we need to rearrange the function to solve for $x$:

$$x = \frac{\text{Final Cost}}{0.75}$$

Q: What is the relationship between the price on the tag and the final cost?

A: The relationship between the price on the tag and the final cost is a linear relationship. This means that as the price on the tag increases, the final cost also increases at a constant rate of $75\%$.

Conclusion

In conclusion, the function that represents the situation where the final cost of a sale item is determined by multiplying the price on the tag by $75\%$ is a linear function of the form $y = 0.75x$. Understanding this function is crucial for both customers and businesses to make informed decisions about sales and discounts.

Key Takeaways

• The final cost of a sale item is determined by multiplying the price on the tag by $75\%$.
• The function that represents the situation is a linear function of the form $y = 0.75x$.
• The graph of the function represents the relationship between the price on the tag and the final cost.
• Understanding this function is crucial for both customers and businesses to make informed decisions about sales and discounts.

Further Exploration

For further exploration, you can try the following:

• Graph the function using different scales and axes.
• Find the equation of the function in terms of the final cost ($y$) as the input variable.
• Use the function to calculate the final cost of different items with varying prices on the tag.
• Explore the concept of discounts and how they affect the final cost of an item.