Describe How The Graph Of The Parent Function Y = X Y=\sqrt{x} Y = X Is Transformed When Graphing Y = − 3 X − 6 Y=-3\sqrt{x-6} Y = − 3 X − 6 .1. The Graph Is Translated 6 Units To The Right.2. The Graph Is Reflected Across The X-axis.3. The Graph Is Vertically Stretched

Mar 10, 2025 by ADMIN 275 views

Transformations of the Parent Function $y=\sqrt{x}$

The parent function $y=\sqrt{x}$ is a fundamental function in mathematics, and understanding its transformations is crucial for graphing and analyzing various functions. In this article, we will explore the transformations of the parent function $y=\sqrt{x}$ when graphing $y=-3\sqrt{x-6}$ .

Understanding the Parent Function

The parent function $y=\sqrt{x}$ is a square root function that takes the input $x$ and returns its square root. The graph of this function is a curve that opens upwards, with the vertex at the origin (0,0). The graph has a characteristic "V" shape, with the curve increasing as $x$ increases.

Transformations of the Parent Function

When graphing $y=-3\sqrt{x-6}$ , we can see that the function has undergone several transformations from the parent function $y=\sqrt{x}$ . Let's analyze each transformation:

1. Horizontal Translation

The first transformation is a horizontal translation of 6 units to the right. This means that the graph of $y=\sqrt{x}$ is shifted 6 units to the right, resulting in a new graph that is centered at (6,0). This transformation is achieved by replacing $x$ with $x-6$ in the parent function.

Horizontal Translation

When we replace $x$ with $x-6$ in the parent function, we are essentially shifting the graph 6 units to the right. This is because the term $x-6$ represents a horizontal shift of 6 units to the right. As a result, the vertex of the graph is now located at (6,0), rather than at the origin (0,0).

2. Reflection Across the x-axis

The second transformation is a reflection across the x-axis. This means that the graph of $y=\sqrt{x}$ is flipped upside down, resulting in a new graph that is centered at (6,0) but has a negative y-value. This transformation is achieved by multiplying the parent function by -1.

Reflection Across the x-axis

When we multiply the parent function by -1, we are essentially reflecting the graph across the x-axis. This is because the negative sign in front of the square root function represents a reflection across the x-axis. As a result, the graph is now centered at (6,0) but has a negative y-value.

3. Vertical Stretch

The third transformation is a vertical stretch by a factor of 3. This means that the graph of $y=\sqrt{x}$ is stretched vertically by a factor of 3, resulting in a new graph that is centered at (6,0) but has a larger y-value. This transformation is achieved by multiplying the parent function by 3.

Vertical Stretch

When we multiply the parent function by 3, we are essentially stretching the graph vertically by a factor of 3. This is because the factor of 3 in front of the square root function represents a vertical stretch. As a result, the graph is now centered at (6,0) but has a larger y-value.

Conclusion

In conclusion, the graph of the parent function $y=\sqrt{x}$ is transformed when graphing $y=-3\sqrt{x-6}$ . The transformations include a horizontal translation of 6 units to the right, a reflection across the x-axis, and a vertical stretch by a factor of 3. Understanding these transformations is crucial for graphing and analyzing various functions in mathematics.

Key Takeaways

The graph of $y=\sqrt{x}$ is translated 6 units to the right when graphing $y=-3\sqrt{x-6}$ .
The graph of $y=\sqrt{x}$ is reflected across the x-axis when graphing $y=-3\sqrt{x-6}$ .
The graph of $y=\sqrt{x}$ is vertically stretched by a factor of 3 when graphing $y=-3\sqrt{x-6}$ .

Final Thoughts

In this article, we have explored the transformations of the parent function $y=\sqrt{x}$ when graphing $y=-3\sqrt{x-6}$ . We have seen that the graph undergoes a horizontal translation, a reflection across the x-axis, and a vertical stretch. Understanding these transformations is crucial for graphing and analyzing various functions in mathematics.
Q&A: Transformations of the Parent Function $y=\sqrt{x}$

In our previous article, we explored the transformations of the parent function $y=\sqrt{x}$ when graphing $y=-3\sqrt{x-6}$ . We saw that the graph undergoes a horizontal translation, a reflection across the x-axis, and a vertical stretch. In this article, we will answer some frequently asked questions about these transformations.

Q: What is the effect of the horizontal translation on the graph of $y=\sqrt{x}$ ?

A: The horizontal translation of 6 units to the right shifts the graph of $y=\sqrt{x}$ to the right, resulting in a new graph that is centered at (6,0). This means that the vertex of the graph is now located at (6,0), rather than at the origin (0,0).

Q: How does the reflection across the x-axis affect the graph of $y=\sqrt{x}$ ?

A: The reflection across the x-axis flips the graph of $y=\sqrt{x}$ upside down, resulting in a new graph that is centered at (6,0) but has a negative y-value. This means that the graph is now centered at (6,0) but has a negative y-value.

Q: What is the effect of the vertical stretch on the graph of $y=\sqrt{x}$ ?

A: The vertical stretch by a factor of 3 stretches the graph of $y=\sqrt{x}$ vertically, resulting in a new graph that is centered at (6,0) but has a larger y-value. This means that the graph is now centered at (6,0) but has a larger y-value.

Q: How do the transformations of the parent function $y=\sqrt{x}$ affect the graph of $y=-3\sqrt{x-6}$ ?

A: The transformations of the parent function $y=\sqrt{x}$ result in a graph of $y=-3\sqrt{x-6}$ that is translated 6 units to the right, reflected across the x-axis, and vertically stretched by a factor of 3.

Q: Can you provide an example of how to graph the function $y=-3\sqrt{x-6}$ ?

A: To graph the function $y=-3\sqrt{x-6}$ , start by graphing the parent function $y=\sqrt{x}$ . Then, apply the transformations of horizontal translation, reflection across the x-axis, and vertical stretch to obtain the graph of $y=-3\sqrt{x-6}$ .

Q: How do the transformations of the parent function $y=\sqrt{x}$ affect the domain and range of the graph of $y=-3\sqrt{x-6}$ ?

A: The transformations of the parent function $y=\sqrt{x}$ result in a graph of $y=-3\sqrt{x-6}$ that has a domain of $x \geq 6$ and a range of $y \leq 0$ .

Q: Can you provide a summary of the transformations of the parent function $y=\sqrt{x}$ ?

A: The transformations of the parent function $y=\sqrt{x}$ include a horizontal translation of 6 units to the right, a reflection across the x-axis, and a vertical stretch by a factor of 3. These transformations result in a graph of $y=-3\sqrt{x-6}$ that is centered at (6,0) but has a negative y-value and a larger y-value.

Conclusion

In this article, we have answered some frequently asked questions about the transformations of the parent function $y=\sqrt{x}$ . We have seen that the graph undergoes a horizontal translation, a reflection across the x-axis, and a vertical stretch. Understanding these transformations is crucial for graphing and analyzing various functions in mathematics.