Juno's Taxi Ride Determining Rate Of Change In A Linear Function

by Admin 65 views

In this article, we will delve into a mathematical problem involving Juno's taxi ride. The problem presents a table representing a linear function, showcasing the amount she owed after traveling various distances. Our main focus will be on determining the rate of change in this scenario. Specifically, we will investigate whether the rate of change is indeed $2.25.

Understanding Linear Functions and Rate of Change

Before diving into the specifics of Juno's taxi ride, let's first establish a solid understanding of the fundamental concepts at play: linear functions and rate of change. This foundational knowledge will be crucial in accurately analyzing the given data and arriving at a well-supported conclusion.

A linear function is a mathematical relationship between two variables that can be represented by a straight line on a graph. This straight line is characterized by a constant rate of change, which essentially describes how much the dependent variable (typically represented on the y-axis) changes for every unit increase in the independent variable (typically represented on the x-axis). In simpler terms, the rate of change tells us how quickly the output of the function changes in response to changes in the input.

The rate of change is often referred to as the slope of the line. A positive rate of change indicates that the dependent variable increases as the independent variable increases, resulting in an upward-sloping line. Conversely, a negative rate of change implies that the dependent variable decreases as the independent variable increases, leading to a downward-sloping line. A rate of change of zero signifies a horizontal line, indicating no change in the dependent variable regardless of changes in the independent variable.

The rate of change can be calculated using the following formula:

Rate of Change = (Change in Dependent Variable) / (Change in Independent Variable)

In the context of a graph, this translates to:

Rate of Change = (Change in y) / (Change in x)

Understanding these concepts is essential for interpreting and analyzing real-world scenarios that can be modeled using linear functions, such as the cost of a taxi ride based on the distance traveled, which is precisely the scenario we will explore in Juno's case. By grasping the relationship between linear functions, rate of change, and their graphical representation, we can confidently tackle the problem at hand and determine whether the rate of change is indeed $2.25.

Analyzing the Table: Miles Traveled vs. Amount Owed

Now, let's turn our attention to the specific data provided in the table. The table presents a clear relationship between the number of miles Juno traveled in the taxi and the corresponding amount she owed. This data forms the basis of our analysis, allowing us to calculate the rate of change and ultimately determine if it matches the proposed value of $2.25.

The table displays two key variables: Miles traveled, which serves as the independent variable (x), and Amount Owed in dollars, which acts as the dependent variable (y). Each row in the table represents a data point, providing us with a specific pair of values for miles traveled and the corresponding amount owed. By examining these data points, we can discern the pattern in how the amount owed changes as the distance traveled increases.

To effectively analyze the data, we can select any two distinct data points from the table. These data points will provide us with the necessary information to calculate the rate of change. Let's denote the two selected data points as (x1, y1) and (x2, y2). Once we have these data points, we can apply the formula for calculating the rate of change:

Rate of Change = (y2 - y1) / (x2 - x1)

By substituting the values from our chosen data points into this formula, we will obtain a numerical value representing the rate of change. This value will tell us how much the amount owed changes for each additional mile traveled. It is crucial to perform this calculation accurately, as it forms the core of our analysis and will ultimately determine whether the rate of change is indeed $2.25.

Furthermore, it is important to note that since the problem states that the table represents a linear function, we should expect to obtain the same rate of change regardless of which two data points we select from the table. This consistency in the rate of change is a hallmark of linear relationships and reinforces the validity of our analysis. If we were to calculate different rates of change using different data points, it would suggest that the relationship is not linear, and our initial assumption would be incorrect. Therefore, we can use this principle as a check on our calculations and ensure the accuracy of our results.

Calculating the Rate of Change

To determine the rate of change in Juno's taxi fare, we will now perform the calculations using the data provided in the table. As discussed earlier, we need to select two distinct data points from the table and apply the rate of change formula. Let's choose the first two data points for this calculation:

  • Data Point 1: (1 mile, $2.50)
  • Data Point 2: (2 miles, $4.75)

Here, x1 = 1, y1 = 2.50, x2 = 2, and y2 = 4.75. Now, we can substitute these values into the rate of change formula:

Rate of Change = (y2 - y1) / (x2 - x1)

Rate of Change = (4.75 - 2.50) / (2 - 1)

Rate of Change = 2.25 / 1

Rate of Change = 2.25

The result of our calculation shows that the rate of change is indeed $2.25 per mile. This means that for every additional mile Juno travels in the taxi, the amount she owes increases by $2.25. This value represents the cost per mile, which is a crucial parameter in understanding the taxi fare structure.

To further validate our result and ensure the accuracy of our calculations, we can repeat the calculation using a different pair of data points from the table. This will help us confirm that the rate of change remains constant, which is a characteristic of linear functions. If we obtain the same rate of change using different data points, it strengthens our conclusion that the relationship between miles traveled and amount owed is indeed linear, and the rate of change is consistently $2.25 per mile. This thorough approach to calculation and verification enhances the reliability of our analysis and provides a solid foundation for our final answer.

Verifying the Rate of Change with Another Data Point

To ensure the consistency and accuracy of our calculated rate of change, we will now repeat the calculation using a different pair of data points from the table. This step is crucial in verifying that the relationship between miles traveled and amount owed is indeed linear and that the rate of change remains constant throughout the journey.

Let's select the following data points for this verification:

  • Data Point 1: (3 miles, $7.00)
  • Data Point 2: (4 miles, $9.25)

Here, x1 = 3, y1 = 7.00, x2 = 4, and y2 = 9.25. Substituting these values into the rate of change formula:

Rate of Change = (y2 - y1) / (x2 - x1)

Rate of Change = (9.25 - 7.00) / (4 - 3)

Rate of Change = 2.25 / 1

Rate of Change = 2.25

As we can see, the rate of change calculated using this new pair of data points is also $2.25 per mile. This result confirms our earlier finding and reinforces the conclusion that the relationship between miles traveled and amount owed is linear, with a constant rate of change. The consistency in the rate of change across different segments of the journey provides strong evidence that the taxi fare structure follows a linear model.

This verification step is essential in ensuring the robustness of our analysis. By demonstrating that the rate of change remains the same regardless of the data points used in the calculation, we can confidently assert that our result is not merely a coincidence but rather a reflection of the underlying linear relationship. This thorough approach to verification enhances the reliability of our findings and allows us to draw a well-supported conclusion regarding the rate of change in Juno's taxi fare.

Conclusion: Is the Rate of Change $2.25?

Based on our detailed analysis and calculations, we can confidently conclude that the rate of change in Juno's taxi fare is indeed $2.25 per mile. This conclusion is supported by the following key findings:

  • We calculated the rate of change using two different pairs of data points from the table.
  • In both calculations, we obtained the same rate of change of $2.25 per mile.
  • The consistency in the rate of change confirms that the relationship between miles traveled and amount owed is linear.

Therefore, the answer to the question