To Graph The Square Root Function $y=\frac{1}{3} \sqrt{x-3}+2$, Complete The Table Of Data Points For The Function. $\[ \begin{array}{|c|c|} \hline x & Y \\ \hline 3 & 2.0 \\ \hline 4 & \square \\ \hline 7 & \square

Introduction

Graphing a square root function can be a challenging task, especially when dealing with functions that involve variables within the square root. In this article, we will focus on completing the table of data points for the function $y=\frac{1}{3} \sqrt{x-3}+2$. This function is a square root function that has been shifted and scaled to create a unique graph.

Understanding the Function

Before we can complete the table of data points, we need to understand the function itself. The function $y=\frac{1}{3} \sqrt{x-3}+2$ is a square root function that has been shifted 3 units to the right and scaled by a factor of $\frac{1}{3}$. The function also has a vertical shift of 2 units.

Completing the Table of Data Points

To complete the table of data points, we need to plug in different values of $x$ into the function and calculate the corresponding values of $y$. We will start by plugging in the values of $x$ that are given in the table.

$x$	$y$
3	2.0
4
7

Calculating the Value of $y$ for $x=4$

To calculate the value of $y$ for $x=4$, we need to plug in $x=4$ into the function and simplify.

$$y=\frac{1}{3} \sqrt{4-3}+2$$

$$y=\frac{1}{3} \sqrt{1}+2$$

$$y=\frac{1}{3} (1)+2$$

$$y=\frac{1}{3} +2$$

$$y=\frac{1}{3} +\frac{6}{3}$$

$$y=\frac{7}{3}$$

So, the value of $y$ for $x=4$ is $\frac{7}{3}$.

Calculating the Value of $y$ for $x=7$

To calculate the value of $y$ for $x=7$, we need to plug in $x=7$ into the function and simplify.

$$y=\frac{1}{3} \sqrt{7-3}+2$$

$$y=\frac{1}{3} \sqrt{4}+2$$

$$y=\frac{1}{3} (2)+2$$

$$y=\frac{2}{3} +2$$

$$y=\frac{2}{3} +\frac{6}{3}$$

$$y=\frac{8}{3}$$

So, the value of $y$ for $x=7$ is $\frac{8}{3}$.

Completed Table of Data Points

Now that we have calculated the values of $y$ for $x=4$ and $x=7$, we can complete the table of data points.

$x$	$y$
3	2.0
4	$\frac{7}{3}$
7	$\frac{8}{3}$

Graphing the Function

Now that we have completed the table of data points, we can graph the function. To graph the function, we need to plot the points on a coordinate plane and draw a smooth curve through the points.

Conclusion

In this article, we completed the table of data points for the function $y=\frac{1}{3} \sqrt{x-3}+2$. We calculated the values of $y$ for $x=4$ and $x=7$ and completed the table of data points. We also graphed the function by plotting the points on a coordinate plane and drawing a smooth curve through the points. This article provides a step-by-step guide on how to complete the table of data points and graph a square root function.

Discussion

The function $y=\frac{1}{3} \sqrt{x-3}+2$ is a square root function that has been shifted and scaled to create a unique graph. The function has a vertical shift of 2 units and a horizontal shift of 3 units to the right. The function also has a scaling factor of $\frac{1}{3}$.

The table of data points provides a way to visualize the function and understand its behavior. By completing the table of data points, we can see how the function changes as the input value $x$ changes.

The graph of the function provides a visual representation of the function and its behavior. By graphing the function, we can see how the function changes as the input value $x$ changes.

References

• [1] "Graphing Square Root Functions" by Math Open Reference
• [2] "Square Root Functions" by Khan Academy

Keywords

• Square root function
• Graphing functions
• Table of data points
• Vertical shift
• Horizontal shift
• Scaling factor
• Coordinate plane
• Graphing functions
• Mathematics
  Frequently Asked Questions: Completing the Table of Data Points for the Square Root Function =====================================================================================

Introduction

In our previous article, we completed the table of data points for the function $y=\frac{1}{3} \sqrt{x-3}+2$. In this article, we will answer some frequently asked questions about completing the table of data points for the square root function.

Q: What is the purpose of completing the table of data points for the square root function?

A: The purpose of completing the table of data points for the square root function is to visualize the function and understand its behavior. By completing the table of data points, we can see how the function changes as the input value $x$ changes.

Q: How do I calculate the value of $y$ for a given value of $x$ in the square root function?

A: To calculate the value of $y$ for a given value of $x$ in the square root function, you need to plug in the value of $x$ into the function and simplify. For example, if we want to calculate the value of $y$ for $x=4$, we need to plug in $x=4$ into the function and simplify.

Q: What is the difference between a vertical shift and a horizontal shift in the square root function?

A: A vertical shift is a shift in the $y$-direction, while a horizontal shift is a shift in the $x$-direction. In the function $y=\frac{1}{3} \sqrt{x-3}+2$, the vertical shift is 2 units, and the horizontal shift is 3 units to the right.

Q: How do I graph the square root function using the completed table of data points?

A: To graph the square root function using the completed table of data points, you need to plot the points on a coordinate plane and draw a smooth curve through the points.

Q: What is the significance of the scaling factor in the square root function?

A: The scaling factor in the square root function determines how much the function is stretched or compressed. In the function $y=\frac{1}{3} \sqrt{x-3}+2$, the scaling factor is $\frac{1}{3}$, which means that the function is compressed by a factor of 3.

Q: Can I use the completed table of data points to find the domain and range of the square root function?

A: Yes, you can use the completed table of data points to find the domain and range of the square root function. By looking at the table of data points, you can see that the domain of the function is all real numbers greater than or equal to 3, and the range of the function is all real numbers greater than or equal to 2.

Q: How do I determine the type of function that I am dealing with?

A: To determine the type of function that you are dealing with, you need to look at the function and identify its characteristics. In the case of the square root function, you can see that it has a square root term, which indicates that it is a square root function.

Q: Can I use the completed table of data points to find the equation of the square root function?

A: Yes, you can use the completed table of data points to find the equation of the square root function. By looking at the table of data points, you can see that the equation of the function is $y=\frac{1}{3} \sqrt{x-3}+2$.

Conclusion

In this article, we answered some frequently asked questions about completing the table of data points for the square root function. We discussed the purpose of completing the table of data points, how to calculate the value of $y$ for a given value of $x$, and how to graph the function using the completed table of data points. We also discussed the significance of the scaling factor and how to determine the type of function that you are dealing with.

Discussion

The square root function is a type of function that has a square root term. It is used to model real-world phenomena that involve square roots, such as the area of a square or the length of a side of a square. The completed table of data points provides a way to visualize the function and understand its behavior.

References

• [1] "Graphing Square Root Functions" by Math Open Reference
• [2] "Square Root Functions" by Khan Academy

Keywords

• Square root function
• Graphing functions
• Table of data points
• Vertical shift
• Horizontal shift
• Scaling factor
• Coordinate plane
• Graphing functions
• Mathematics