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Introduction

In geometry, we often encounter linear patterns that can be represented algebraically. A table showing a linear pattern can be used to derive a function that relates the variables $x$ and $y$. In this article, we will explore the table, identify the variables $x$ and $y$, and write a function that relates $y$ to $x$.

The Table

The table below shows a familiar linear pattern from geometry.

$x$	$y$
1	3
2	5

Understanding the Variables

The variables $x$ and $y$ represent the input and output values of a linear function. In this case, the table suggests that the output value $y$ is increasing by 2 units for every 1 unit increase in the input value $x$.

Writing the Function

To write a function that relates $y$ to $x$, we need to identify the slope and the y-intercept of the linear function. The slope represents the rate of change of the output value with respect to the input value, while the y-intercept represents the value of the output when the input is 0.

Calculating the Slope

The slope of the linear function can be calculated using the formula:

$$m = \frac{\Delta y}{\Delta x}$$

where $\Delta y$ is the change in the output value and $\Delta x$ is the change in the input value.

In this case, the change in the output value is 2 units (from 3 to 5), and the change in the input value is 1 unit (from 1 to 2). Therefore, the slope is:

$$m = \frac{2}{1} = 2$$

Calculating the Y-Intercept

The y-intercept of the linear function can be calculated using the formula:

$$b = y - mx$$

where $y$ is the output value when the input is 0, $m$ is the slope, and $x$ is the input value.

In this case, the output value when the input is 0 is not given in the table. However, we can use the point-slope form of a linear equation to write the equation of the line:

$$y - y_1 = m(x - x_1)$$

where $(x_1, y_1)$ is a point on the line.

Using the point $(1, 3)$, we can write the equation of the line as:

$$y - 3 = 2(x - 1)$$

Simplifying the equation, we get:

$$y = 2x + 1$$

Conclusion

In this article, we have explored a table showing a familiar linear pattern from geometry. We have identified the variables $x$ and $y$, and written a function that relates $y$ to $x$. The function is a linear equation of the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

The Variables $x$ and $y$ Represent

The variables $x$ and $y$ represent the input and output values of a linear function. In this case, the input value $x$ represents the number of units, and the output value $y$ represents the total number of units.

The Function Relates $y$ to $x$

The function $y = 2x + 1$ relates the output value $y$ to the input value $x$. The function is a linear equation that describes the relationship between the input and output values.

The Slope and Y-Intercept

The slope of the linear function is 2, which represents the rate of change of the output value with respect to the input value. The y-intercept of the linear function is 1, which represents the value of the output when the input is 0.

The Point-Slope Form

The point-slope form of a linear equation is $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line. Using the point $(1, 3)$, we can write the equation of the line as $y - 3 = 2(x - 1)$.

Simplifying the Equation

Simplifying the equation $y - 3 = 2(x - 1)$, we get $y = 2x + 1$. This is the final equation of the line.

The Final Equation

The final equation of the line is $y = 2x + 1$. This equation describes the relationship between the input and output values of the linear function.

The Linear Function

The linear function is $y = 2x + 1$. This function describes the relationship between the input and output values of the linear function.

The Graph of the Linear Function

The graph of the linear function is a straight line with a slope of 2 and a y-intercept of 1.

The Linear Pattern

The linear pattern is a straight line with a slope of 2 and a y-intercept of 1.

The Table Shows a Familiar Linear Pattern

The table shows a familiar linear pattern from geometry.

The Variables $x$ and $y$ Represent

The variables $x$ and $y$ represent the input and output values of a linear function.

The Function Relates $y$ to $x$

The function $y = 2x + 1$ relates the output value $y$ to the input value $x$.

The Slope and Y-Intercept

The slope of the linear function is 2, which represents the rate of change of the output value with respect to the input value. The y-intercept of the linear function is 1, which represents the value of the output when the input is 0.

The Point-Slope Form

The point-slope form of a linear equation is $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line.

Simplifying the Equation

Simplifying the equation $y - 3 = 2(x - 1)$, we get $y = 2x + 1$. This is the final equation of the line.

The Final Equation

The final equation of the line is $y = 2x + 1$. This equation describes the relationship between the input and output values of the linear function.

The Linear Function

The linear function is $y = 2x + 1$. This function describes the relationship between the input and output values of the linear function.

The Graph of the Linear Function

The graph of the linear function is a straight line with a slope of 2 and a y-intercept of 1.

The Linear Pattern

The linear pattern is a straight line with a slope of 2 and a y-intercept of 1.

Conclusion

$$$$$$$$$$$$$$

Q: What is the table showing a familiar linear pattern from geometry?

A: The table is showing a linear pattern where the output value $y$ is increasing by 2 units for every 1 unit increase in the input value $x$.

Q: What do the variables $x$ and $y$ represent?

A: The variables $x$ and $y$ represent the input and output values of a linear function. In this case, the input value $x$ represents the number of units, and the output value $y$ represents the total number of units.

Q: How do we write a function that relates $y$ to $x$?

A: To write a function that relates $y$ to $x$, we need to identify the slope and the y-intercept of the linear function. The slope represents the rate of change of the output value with respect to the input value, while the y-intercept represents the value of the output when the input is 0.

Q: How do we calculate the slope of the linear function?

A: The slope of the linear function can be calculated using the formula:

$$m = \frac{\Delta y}{\Delta x}$$

where $\Delta y$ is the change in the output value and $\Delta x$ is the change in the input value.

Q: How do we calculate the y-intercept of the linear function?

A: The y-intercept of the linear function can be calculated using the formula:

$$b = y - mx$$

where $y$ is the output value when the input is 0, $m$ is the slope, and $x$ is the input value.

Q: What is the point-slope form of a linear equation?

A: The point-slope form of a linear equation is:

$$y - y_1 = m(x - x_1)$$

where $(x_1, y_1)$ is a point on the line.

Q: How do we simplify the equation of the line?

A: To simplify the equation of the line, we can use the point-slope form and substitute the values of $x_1$ and $y_1$.

Q: What is the final equation of the line?

A: The final equation of the line is:

$$y = 2x + 1$$

This equation describes the relationship between the input and output values of the linear function.

Q: What is the graph of the linear function?

A: The graph of the linear function is a straight line with a slope of 2 and a y-intercept of 1.

Q: What is the linear pattern?

A: The linear pattern is a straight line with a slope of 2 and a y-intercept of 1.

Q: What is the table showing a familiar linear pattern from geometry?

A: The table is showing a linear pattern where the output value $y$ is increasing by 2 units for every 1 unit increase in the input value $x$.

Q: What do the variables $x$ and $y$ represent?

A: The variables $x$ and $y$ represent the input and output values of a linear function. In this case, the input value $x$ represents the number of units, and the output value $y$ represents the total number of units.

Q: How do we write a function that relates $y$ to $x$?

A: To write a function that relates $y$ to $x$, we need to identify the slope and the y-intercept of the linear function. The slope represents the rate of change of the output value with respect to the input value, while the y-intercept represents the value of the output when the input is 0.

Q: How do we calculate the slope of the linear function?

A: The slope of the linear function can be calculated using the formula:

$$m = \frac{\Delta y}{\Delta x}$$

where $\Delta y$ is the change in the output value and $\Delta x$ is the change in the input value.

Q: How do we calculate the y-intercept of the linear function?

A: The y-intercept of the linear function can be calculated using the formula:

$$b = y - mx$$

where $y$ is the output value when the input is 0, $m$ is the slope, and $x$ is the input value.

Q: What is the point-slope form of a linear equation?

A: The point-slope form of a linear equation is:

$$y - y_1 = m(x - x_1)$$

where $(x_1, y_1)$ is a point on the line.

Q: How do we simplify the equation of the line?

A: To simplify the equation of the line, we can use the point-slope form and substitute the values of $x_1$ and $y_1$.

Q: What is the final equation of the line?

A: The final equation of the line is:

$$y = 2x + 1$$

This equation describes the relationship between the input and output values of the linear function.

Q: What is the graph of the linear function?

A: The graph of the linear function is a straight line with a slope of 2 and a y-intercept of 1.

Q: What is the linear pattern?

A: The linear pattern is a straight line with a slope of 2 and a y-intercept of 1.

Conclusion

In this Q&A article, we have explored the table showing a familiar linear pattern from geometry. We have identified the variables $x$ and $y$, and written a function that relates $y$ to $x$. The function is a linear equation of the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.