The Table Below Represents A Function.$\[ \begin{tabular}{|l|c|c|c|c|c|} \hline $x$ & 1 & 2 & 3 & 4 & 5 \\ \hline $y$ & 6 & 12 & 18 & 24 & 30 \\ \hline \end{tabular} \\]Which Statement Would Best Describe The Graph Of The Function?A. The Graph

Introduction

In mathematics, a function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the range. The table below represents a function, where each input value of $x$ corresponds to a unique output value of $y$. In this article, we will explore the graph of the function represented by the table and determine which statement best describes it.

The Table

$x$	1	2	3	4	5
$y$	6	12	18	24	30

Understanding the Graph

To understand the graph of the function, we need to analyze the relationship between the input values of $x$ and the output values of $y$. From the table, we can see that each input value of $x$ corresponds to a unique output value of $y$. This means that the function is a one-to-one function, where each input value maps to a unique output value.

Linear Function

The graph of the function represented by the table appears to be a straight line. This is because the output values of $y$ increase by a constant amount for each increase in the input values of $x$. Specifically, the output values of $y$ increase by 6 for each increase in the input values of $x$ by 1.

Equation of the Line

To find the equation of the line, we need to determine the slope and the y-intercept. The slope is the change in the output values of $y$ divided by the change in the input values of $x$. In this case, the slope is 6, since the output values of $y$ increase by 6 for each increase in the input values of $x$ by 1.

The y-intercept is the value of $y$ when $x$ is equal to 0. However, since the table only represents values of $x$ from 1 to 5, we cannot determine the y-intercept directly from the table. Nevertheless, we can still write the equation of the line in the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Equation of the Line: $y = 6x + b$

Since the slope is 6, we can write the equation of the line as $y = 6x + b$. To find the value of $b$, we need to determine the y-intercept. However, as mentioned earlier, we cannot determine the y-intercept directly from the table.

Finding the Y-Intercept

To find the y-intercept, we need to find the value of $y$ when $x$ is equal to 0. However, since the table only represents values of $x$ from 1 to 5, we cannot determine the y-intercept directly from the table. Nevertheless, we can still use the equation of the line to find the y-intercept.

Finding the Y-Intercept: $b = -3$

To find the y-intercept, we need to substitute $x = 0$ into the equation of the line. This gives us:

$y = 6(0) + b$ $y = b$

Since the y-intercept is the value of $y$ when $x$ is equal to 0, we can set $y$ equal to 0 and solve for $b$:

$0 = b$ $b = -3$

Therefore, the y-intercept is $-3$.

Equation of the Line: $y = 6x - 3$

Now that we have found the y-intercept, we can write the equation of the line as $y = 6x - 3$.

Conclusion

In conclusion, the graph of the function represented by the table is a straight line with a slope of 6 and a y-intercept of $-3$. The equation of the line is $y = 6x - 3$. This means that for every increase in the input values of $x$ by 1, the output values of $y$ increase by 6.

Discussion

The graph of the function represented by the table is a classic example of a linear function. Linear functions are characterized by a constant rate of change, which is the slope of the line. In this case, the slope is 6, which means that the output values of $y$ increase by 6 for each increase in the input values of $x$ by 1.

Real-World Applications

Linear functions have many real-world applications, including modeling population growth, predicting stock prices, and calculating the cost of goods. In each of these cases, the linear function can be used to make predictions and forecasts based on the input values.

Conclusion

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Q: What is the relationship between the input values of $x$ and the output values of $y$ in the table?

A: The relationship between the input values of $x$ and the output values of $y$ in the table is a one-to-one function, where each input value maps to a unique output value.

Q: What type of function is represented by the table?

A: The function represented by the table is a linear function, characterized by a constant rate of change.

Q: What is the slope of the line represented by the table?

A: The slope of the line represented by the table is 6, which means that the output values of $y$ increase by 6 for each increase in the input values of $x$ by 1.

Q: What is the y-intercept of the line represented by the table?

A: The y-intercept of the line represented by the table is $-3$, which means that the value of $y$ when $x$ is equal to 0 is $-3$.

Q: What is the equation of the line represented by the table?

A: The equation of the line represented by the table is $y = 6x - 3$.

Q: How can the equation of the line be used in real-world applications?

A: The equation of the line can be used in real-world applications such as modeling population growth, predicting stock prices, and calculating the cost of goods.

Q: What are some common characteristics of linear functions?

A: Some common characteristics of linear functions include a constant rate of change, a straight line graph, and an equation of the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Q: How can the table be used to determine the equation of the line?

A: The table can be used to determine the equation of the line by analyzing the relationship between the input values of $x$ and the output values of $y$. By identifying the slope and the y-intercept, the equation of the line can be written in the form $y = mx + b$.

Q: What are some potential limitations of using the table to determine the equation of the line?

A: Some potential limitations of using the table to determine the equation of the line include the fact that the table only represents values of $x$ from 1 to 5, and the y-intercept cannot be determined directly from the table.

Q: How can the equation of the line be used to make predictions and forecasts?

A: The equation of the line can be used to make predictions and forecasts by substituting different values of $x$ into the equation and solving for $y$. This can be useful in a variety of real-world applications, such as modeling population growth or predicting stock prices.

Q: What are some potential applications of linear functions in real-world scenarios?

A: Some potential applications of linear functions in real-world scenarios include:

• Modeling population growth
• Predicting stock prices
• Calculating the cost of goods
• Determining the rate of change of a quantity
• Making predictions and forecasts based on historical data

Q: How can linear functions be used to solve problems in real-world scenarios?

A: Linear functions can be used to solve problems in real-world scenarios by analyzing the relationship between the input values and the output values, identifying the slope and the y-intercept, and using the equation of the line to make predictions and forecasts.