Describe The Transformation Of The Graph Of The Parent Function $y=\sqrt{x}$ For The Function $y=\sqrt{x+7}+5$.The Graph Is Shifted:- 7 Units Left- 5 Units UpWhat Is The Domain Of $y=\sqrt{x+7}+5$?A. $x \geq 0$B.

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Introduction

The parent function $y=\sqrt{x}$ is a fundamental function in mathematics, and its graph is a fundamental concept in algebra and geometry. The graph of the parent function $y=\sqrt{x}$ is a half-parabola that opens upwards, with its vertex at the origin (0, 0). In this article, we will discuss the transformation of the graph of the parent function $y=\sqrt{x}$ for the function $y=\sqrt{x+7}+5$. We will also determine the domain of the function $y=\sqrt{x+7}+5$.

Transformation of the Graph

The function $y=\sqrt{x+7}+5$ is a transformation of the parent function $y=\sqrt{x}$. To understand the transformation, let's analyze the function. The function $y=\sqrt{x+7}+5$ can be broken down into two parts: $\sqrt{x+7}$ and $+5$. The first part, $\sqrt{x+7}$, is a horizontal shift of the parent function $y=\sqrt{x}$ by 7 units to the left. This means that the graph of the function $y=\sqrt{x+7}$ is the same as the graph of the parent function $y=\sqrt{x}$, but shifted 7 units to the left.

The second part, $+5$, is a vertical shift of the graph of the function $y=\sqrt{x+7}$ by 5 units upwards. This means that the graph of the function $y=\sqrt{x+7}+5$ is the same as the graph of the function $y=\sqrt{x+7}$, but shifted 5 units upwards.

Shifting the Graph

The graph of the function $y=\sqrt{x+7}+5$ is shifted 7 units to the left and 5 units upwards. This means that the vertex of the graph is at the point (-7, 5). The graph is also reflected across the x-axis, since the function is in the form $y=\sqrt{x+7}+5$, which is a positive function.

Domain of the Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. In the case of the function $y=\sqrt{x+7}+5$, the domain is all real numbers x such that $x+7 \geq 0$. This means that the domain of the function is $x \geq -7$.

Conclusion

In conclusion, the graph of the function $y=\sqrt{x+7}+5$ is a transformation of the parent function $y=\sqrt{x}$. The graph is shifted 7 units to the left and 5 units upwards, and the domain of the function is $x \geq -7$. Understanding the transformation of the graph and the domain of the function is essential in algebra and geometry.

Graph of the Function

The graph of the function $y=\sqrt{x+7}+5$ is a half-parabola that opens upwards, with its vertex at the point (-7, 5). The graph is shifted 7 units to the left and 5 units upwards, and it is reflected across the x-axis.

Domain of the Function

The domain of the function $y=\sqrt{x+7}+5$ is $x \geq -7$. This means that the function is defined for all real numbers x such that $x+7 \geq 0$.

Transformation of the Graph

The graph of the function $y=\sqrt{x+7}+5$ is a transformation of the parent function $y=\sqrt{x}$. The graph is shifted 7 units to the left and 5 units upwards, and it is reflected across the x-axis.

Vertical and Horizontal Shifts

The graph of the function $y=\sqrt{x+7}+5$ is a vertical shift of the graph of the function $y=\sqrt{x+7}$ by 5 units upwards. The graph is also a horizontal shift of the graph of the parent function $y=\sqrt{x}$ by 7 units to the left.

Reflection Across the X-Axis

The graph of the function $y=\sqrt{x+7}+5$ is reflected across the x-axis, since the function is in the form $y=\sqrt{x+7}+5$, which is a positive function.

Domain of the Function

The domain of the function $y=\sqrt{x+7}+5$ is $x \geq -7$. This means that the function is defined for all real numbers x such that $x+7 \geq 0$.

Conclusion

In conclusion, the graph of the function $y=\sqrt{x+7}+5$ is a transformation of the parent function $y=\sqrt{x}$. The graph is shifted 7 units to the left and 5 units upwards, and the domain of the function is $x \geq -7$. Understanding the transformation of the graph and the domain of the function is essential in algebra and geometry.

Final Thoughts

The transformation of the graph of the parent function $y=\sqrt{x}$ for the function $y=\sqrt{x+7}+5$ is a fundamental concept in algebra and geometry. Understanding the transformation of the graph and the domain of the function is essential in mathematics. The graph of the function $y=\sqrt{x+7}+5$ is a half-parabola that opens upwards, with its vertex at the point (-7, 5). The domain of the function is $x \geq -7$.

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Q: What is the parent function $y=\sqrt{x}$?

A: The parent function $y=\sqrt{x}$ is a fundamental function in mathematics, and its graph is a half-parabola that opens upwards, with its vertex at the origin (0, 0).

Q: What is the transformation of the graph of the parent function $y=\sqrt{x}$ for the function $y=\sqrt{x+7}+5$?

A: The graph of the function $y=\sqrt{x+7}+5$ is a transformation of the parent function $y=\sqrt{x}$. The graph is shifted 7 units to the left and 5 units upwards, and it is reflected across the x-axis.

Q: What is the domain of the function $y=\sqrt{x+7}+5$?

A: The domain of the function $y=\sqrt{x+7}+5$ is $x \geq -7$. This means that the function is defined for all real numbers x such that $x+7 \geq 0$.

Q: How do you determine the domain of a function?

A: To determine the domain of a function, you need to find the values of x for which the function is defined. In the case of the function $y=\sqrt{x+7}+5$, the domain is all real numbers x such that $x+7 \geq 0$.

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

A: A horizontal shift is a shift of the graph of a function to the left or right, while a vertical shift is a shift of the graph of a function up or down.

Q: How do you determine the type of shift (horizontal or vertical) of a function?

A: To determine the type of shift of a function, you need to analyze the function and identify the values that are being added or subtracted from the parent function.

Q: What is the significance of the vertex of a graph?

A: The vertex of a graph is the point at which the graph changes direction. In the case of the graph of the function $y=\sqrt{x+7}+5$, the vertex is at the point (-7, 5).

Q: How do you find the vertex of a graph?

A: To find the vertex of a graph, you need to analyze the function and identify the values of x and y that correspond to the vertex.

Q: What is the relationship between the parent function $y=\sqrt{x}$ and the function $y=\sqrt{x+7}+5$?

A: The function $y=\sqrt{x+7}+5$ is a transformation of the parent function $y=\sqrt{x}$. The graph of the function $y=\sqrt{x+7}+5$ is a shift of the graph of the parent function $y=\sqrt{x}$ by 7 units to the left and 5 units upwards.

Q: How do you determine the type of transformation (shift, reflection, etc.) of a function?

A: To determine the type of transformation of a function, you need to analyze the function and identify the values that are being added or subtracted from the parent function.

Q: What is the significance of the domain of a function?

A: The domain of a function is the set of all possible input values (x-values) for which the function is defined. In the case of the function $y=\sqrt{x+7}+5$, the domain is $x \geq -7$.

Q: How do you determine the domain of a function?

A: To determine the domain of a function, you need to find the values of x for which the function is defined.

Q: What is the relationship between the graph of a function and its domain?

A: The graph of a function is a visual representation of the function, and its domain is the set of all possible input values (x-values) for which the function is defined.

Q: How do you determine the type of graph (linear, quadratic, etc.) of a function?

A: To determine the type of graph of a function, you need to analyze the function and identify the values that are being added or subtracted from the parent function.

Q: What is the significance of the vertex of a graph?

A: The vertex of a graph is the point at which the graph changes direction. In the case of the graph of the function $y=\sqrt{x+7}+5$, the vertex is at the point (-7, 5).

Q: How do you find the vertex of a graph?

A: To find the vertex of a graph, you need to analyze the function and identify the values of x and y that correspond to the vertex.

Q: What is the relationship between the parent function $y=\sqrt{x}$ and the function $y=\sqrt{x+7}+5$?

A: The function $y=\sqrt{x+7}+5$ is a transformation of the parent function $y=\sqrt{x}$. The graph of the function $y=\sqrt{x+7}+5$ is a shift of the graph of the parent function $y=\sqrt{x}$ by 7 units to the left and 5 units upwards.

Q: How do you determine the type of transformation (shift, reflection, etc.) of a function?

A: To determine the type of transformation of a function, you need to analyze the function and identify the values that are being added or subtracted from the parent function.

Q: What is the significance of the domain of a function?

A: The domain of a function is the set of all possible input values (x-values) for which the function is defined. In the case of the function $y=\sqrt{x+7}+5$, the domain is $x \geq -7$.

Q: How do you determine the domain of a function?

A: To determine the domain of a function, you need to find the values of x for which the function is defined.

Q: What is the relationship between the graph of a function and its domain?

A: The graph of a function is a visual representation of the function, and its domain is the set of all possible input values (x-values) for which the function is defined.

Q: How do you determine the type of graph (linear, quadratic, etc.) of a function?

A: To determine the type of graph of a function, you need to analyze the function and identify the values that are being added or subtracted from the parent function.

Q: What is the significance of the vertex of a graph?

A: The vertex of a graph is the point at which the graph changes direction. In the case of the graph of the function $y=\sqrt{x+7}+5$, the vertex is at the point (-7, 5).

Q: How do you find the vertex of a graph?

A: To find the vertex of a graph, you need to analyze the function and identify the values of x and y that correspond to the vertex.

Q: What is the relationship between the parent function $y=\sqrt{x}$ and the function $y=\sqrt{x+7}+5$?

A: The function $y=\sqrt{x+7}+5$ is a transformation of the parent function $y=\sqrt{x}$. The graph of the function $y=\sqrt{x+7}+5$ is a shift of the graph of the parent function $y=\sqrt{x}$ by 7 units to the left and 5 units upwards.

Q: How do you determine the type of transformation (shift, reflection, etc.) of a function?

A: To determine the type of transformation of a function, you need to analyze the function and identify the values that are being added or subtracted from the parent function.

Q: What is the significance of the domain of a function?

A: The domain of a function is the set of all possible input values (x-values) for which the function is defined. In the case of the function $y=\sqrt{x+7}+5$, the domain is $x \geq -7$.

Q: How do you determine the domain of a function?

A: To determine the domain of a function, you need to find the values of x for which the function is defined.

Q: What is the relationship between the graph of a function and its domain?

A: The graph of a function is a visual representation of the function, and its domain is the set of all possible input values (x-values) for which the function is defined.

Q: How do you determine the type of graph (linear, quadratic, etc.) of a function?

A: To determine the type of graph of a function, you need to analyze the function and identify the values that are being added or subtracted from the parent function.

Q: What is the significance of the vertex of a graph?

A: The vertex of a graph is the point at which the graph changes direction. In the case of the graph of the function $y=\sqrt{x+7}+5$, the vertex is at the point (-7, 5).

Q: How do you find the vertex of a graph?

A: To find the vertex of a graph, you need to analyze the function and identify the values