This Is The Input-output Table For The Linear Function $y = 3x$.$\[ \begin{tabular}{|r|r|} \hline $x$ & $y$ \\ \hline -2 & -6 \\ \hline -1 & -3 \\ \hline 0 & 0 \\ \hline 1 & 3 \\ \hline 2 & 6 \\ \hline 3 & 9

Understanding the Input-Output Table for the Linear Function $y = 3x$

In mathematics, a linear function is a function that can be written in the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. The input-output table is a table that shows the input values of the function and the corresponding output values. In this article, we will discuss the input-output table for the linear function $y = 3x$.

What is a Linear Function?

A linear function is a function that can be written in the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. The slope of a linear function is a measure of how much the function changes as the input changes. The y-intercept of a linear function is the point where the function intersects the y-axis.

The Input-Output Table for the Linear Function $y = 3x$

The input-output table for the linear function $y = 3x$ is a table that shows the input values of the function and the corresponding output values. The table is as follows:

How to Read the Input-Output Table

To read the input-output table, we need to understand that the input values are the values of $x$ and the output values are the values of $y$. For example, when $x = -2$, the output value is $y = -6$. This means that when the input value is -2, the output value is -6.

Understanding the Relationship Between Input and Output

From the input-output table, we can see that the output value is always three times the input value. For example, when $x = 1$, the output value is $y = 3$. This means that when the input value is 1, the output value is 3. This relationship is a key characteristic of the linear function $y = 3x$.

How to Use the Input-Output Table

The input-output table can be used to find the output value for a given input value. For example, if we want to find the output value when $x = 4$, we can look at the table and see that the output value is $y = 12$. This means that when the input value is 4, the output value is 12.

In conclusion, the input-output table for the linear function $y = 3x$ is a table that shows the input values of the function and the corresponding output values. The table can be used to understand the relationship between the input and output values of the function. By analyzing the table, we can see that the output value is always three times the input value. This relationship is a key characteristic of the linear function $y = 3x$.

Applications of the Linear Function $y = 3x$

The linear function $y = 3x$ has many applications in real-world problems. For example, it can be used to model the cost of producing a product, where the cost is directly proportional to the number of units produced. It can also be used to model the revenue of a business, where the revenue is directly proportional to the number of units sold.

Real-World Examples of the Linear Function $y = 3x$

Here are some real-world examples of the linear function $y = 3x$:

• Cost of producing a product: Suppose a company produces a product that costs $3 to make each unit. If the company produces 10 units, the total cost will be $30.
• Revenue of a business: Suppose a business sells a product that costs $3 to make each unit. If the business sells 10 units, the total revenue will be $30.
• Distance traveled: Suppose a car travels at a speed of 3 miles per hour. If the car travels for 10 hours, the total distance traveled will be 30 miles.

Solving Problems Using the Linear Function $y = 3x$

Here are some problems that can be solved using the linear function $y = 3x$:

• Problem 1: If a company produces a product that costs $3 to make each unit, and the company produces 20 units, what is the total cost?
• Problem 2: If a business sells a product that costs $3 to make each unit, and the business sells 20 units, what is the total revenue?
• Problem 3: If a car travels at a speed of 3 miles per hour, and the car travels for 20 hours, what is the total distance traveled?

Answer to Problem 1

To solve problem 1, we can use the linear function $y = 3x$. We know that the cost of producing each unit is $3, and the company produces 20 units. We can plug in the values into the function to get:

$y = 3x$ $y = 3(20)$ $y = 60$

Therefore, the total cost is $60.

Answer to Problem 2

To solve problem 2, we can use the linear function $y = 3x$. We know that the cost of producing each unit is $3, and the business sells 20 units. We can plug in the values into the function to get:

$y = 3x$ $y = 3(20)$ $y = 60$

Therefore, the total revenue is $60.

Answer to Problem 3

To solve problem 3, we can use the linear function $y = 3x$. We know that the car travels at a speed of 3 miles per hour, and the car travels for 20 hours. We can plug in the values into the function to get:

$y = 3x$ $y = 3(20)$ $y = 60$

Therefore, the total distance traveled is 60 miles.

In conclusion, the linear function $y = 3x$ has many applications in real-world problems. It can be used to model the cost of producing a product, the revenue of a business, and the distance traveled by a car. By analyzing the input-output table, we can see that the output value is always three times the input value. This relationship is a key characteristic of the linear function $y = 3x$.