Which Description Explains How The Graph Of F ( X ) = X F(x)=\sqrt{x} F ( X ) = X ​ Could Be Transformed To Form The Graph Of G ( X ) = X + 7 G(x)=\sqrt{x+7} G ( X ) = X + 7 ​ ?A. Horizontal Shift Of 7 Units Left B. Vertical Shift Of 7 Units Up C. Horizontal Shift Of 7 Units Right D.

Introduction

In mathematics, graph transformations are essential concepts that help us understand how functions can be manipulated to create new functions. One of the fundamental types of transformations is the horizontal shift, which involves moving the graph of a function to the left or right. In this article, we will explore how the graph of $f(x)=\sqrt{x}$ can be transformed to form the graph of $g(x)=\sqrt{x+7}$.

Understanding the Original Function

The original function is $f(x)=\sqrt{x}$. This function represents a square root function, which is a type of rational function. The graph of this function is a curve that opens upwards, with the x-axis as its asymptote. The function is defined only for non-negative values of x, as the square root of a negative number is not a real number.

Understanding the Target Function

The target function is $g(x)=\sqrt{x+7}$. This function is also a square root function, but it has been modified to include an additional term of 7 inside the square root. This means that the graph of this function will be shifted to the left compared to the graph of the original function.

Analyzing the Transformation

To understand how the graph of $f(x)=\sqrt{x}$ can be transformed to form the graph of $g(x)=\sqrt{x+7}$, we need to analyze the transformation that has been applied. The key difference between the two functions is the additional term of 7 inside the square root of the target function. This term indicates that the graph of the target function has been shifted to the left by 7 units.

Horizontal Shift

A horizontal shift is a type of transformation that involves moving the graph of a function to the left or right. In this case, the graph of $g(x)=\sqrt{x+7}$ has been shifted to the left by 7 units compared to the graph of $f(x)=\sqrt{x}$. This means that for every point (x, y) on the graph of $f(x)=\sqrt{x}$, there is a corresponding point (x-7, y) on the graph of $g(x)=\sqrt{x+7}$.

Conclusion

In conclusion, the graph of $f(x)=\sqrt{x}$ can be transformed to form the graph of $g(x)=\sqrt{x+7}$ by applying a horizontal shift of 7 units to the left. This type of transformation is essential in mathematics, as it helps us understand how functions can be manipulated to create new functions. By analyzing the transformation that has been applied, we can gain a deeper understanding of the properties of the original and target functions.

Answer

The correct answer is:

A. Horizontal shift of 7 units left

Q&A: Transforming Graphs

Q: What is a horizontal shift in graph transformations?

A: A horizontal shift is a type of transformation that involves moving the graph of a function to the left or right. This means that for every point (x, y) on the graph of the original function, there is a corresponding point (x-h, y) on the graph of the transformed function, where h is the horizontal shift.

Q: How do I determine the direction of the horizontal shift?

A: To determine the direction of the horizontal shift, you need to look at the transformation that has been applied to the original function. If the transformation involves adding a constant to the x-value inside the function, the graph will be shifted to the left. If the transformation involves subtracting a constant from the x-value inside the function, the graph will be shifted to the right.

Q: What is the difference between a horizontal shift and a vertical shift?

A: A horizontal shift involves moving the graph of a function to the left or right, while a vertical shift involves moving the graph of a function up or down. A horizontal shift changes the x-value of the points on the graph, while a vertical shift changes the y-value of the points on the graph.

Q: How do I apply a horizontal shift to a function?

A: To apply a horizontal shift to a function, you need to add or subtract a constant from the x-value inside the function. For example, if you want to apply a horizontal shift of 3 units to the left to the function f(x) = x^2, you would get the new function f(x) = (x-3)^2.

Q: What are some common examples of horizontal shifts?

A: Some common examples of horizontal shifts include:

• f(x) = (x-2)^2, which is a horizontal shift of 2 units to the right of the function f(x) = x^2
• f(x) = (x+3)^2, which is a horizontal shift of 3 units to the left of the function f(x) = x^2
• f(x) = (x-5)^2, which is a horizontal shift of 5 units to the right of the function f(x) = x^2

Q: How do I graph a function with a horizontal shift?

A: To graph a function with a horizontal shift, you need to follow these steps:

1. Graph the original function.
2. Identify the direction of the horizontal shift.
3. Move the graph of the original function to the left or right by the specified amount.
4. Label the new graph with the transformed function.

Q: What are some real-world applications of horizontal shifts?

A: Some real-world applications of horizontal shifts include:

• Physics: When an object is moving at a constant velocity, its position can be described using a horizontal shift of the object's position function.
• Engineering: When designing a system, engineers may need to apply horizontal shifts to the system's equations to account for changes in the system's parameters.
• Computer Science: When programming a computer, developers may need to apply horizontal shifts to the program's equations to account for changes in the program's inputs.

Conclusion

In conclusion, horizontal shifts are an essential concept in graph transformations. By understanding how to apply horizontal shifts to functions, you can gain a deeper understanding of the properties of the original and transformed functions. Whether you are a student, a professional, or simply someone interested in mathematics, understanding horizontal shifts can help you to better understand the world around you.