Transformations Of Square Root Functions Analyzing Y=√(-4x-36)

Analyzing the Equation $y=\sqrt{-4x-36}$

Our initial focus should be on the equation $y=\sqrt{-4x-36}$. To decipher the transformations, it's beneficial to rewrite the equation in a more revealing form. Factoring out the constant within the square root helps to clearly identify horizontal transformations. We can rewrite the equation as follows:

$y = \sqrt{-4(x+9)}$

Now, let’s break down each part of the transformation step by step:

1. Horizontal Reflection: The negative sign inside the square root, specifically the “-” in “-4”, indicates a reflection across the y-axis. This is because the input x is being negated, causing the graph to flip horizontally. This reflection is a crucial element in understanding the final orientation of the graph.

2. Horizontal Stretch/Compression: The coefficient “4” inside the square root affects the horizontal scale of the graph. Specifically, the term “-4” implies a horizontal compression by a factor of $\frac{1}{4}$. This might seem counterintuitive, but a larger number inside the function compresses the graph horizontally, while a fraction would stretch it. The horizontal compression is a key aspect of this transformation.

3. Horizontal Translation: The “+9” inside the parenthesis indicates a horizontal translation. The function $y = \sqrt{-4(x+9)}$ is translated 9 units to the left. Remember that transformations inside the function (affecting x) act in the opposite direction to what you might initially expect. Therefore, “+9” moves the graph to the left on the x-axis.

4. Vertical Stretch: Factoring out the 4 gives us $y = \sqrt{-4(x + 9)} = \sqrt{-4} \cdot \sqrt{(x + 9)} = 2i\sqrt{x + 9}$. This implies that there is no vertical stretch factor of 2 in the real plane due to the complex number i. However, since the transformation to be described is in the real plane, there will be no vertical stretch. Instead, there is a horizontal compression by a factor of $\frac{1}{4}$. This is because the square root of 4 is 2, leading to a vertical stretch by a factor of 2. This vertical stretch changes the shape of the graph, making it appear taller compared to the parent function.

Comparing to the Parent Square Root Function $y=\sqrt{x}$

To compare the transformed graph to the parent square root function $y=\sqrt{x}$, we need to summarize the transformations we’ve identified:

1. Reflection over the y-axis: Due to the negative sign inside the square root.
2. Horizontal compression by a factor of $\frac{1}{4}$: Due to the coefficient “4” multiplying x.
3. Horizontal translation 9 units to the left: Due to the “+9” inside the parenthesis.

It is crucial to note that there's no reflection over the x-axis and the translation is to the left, not the right. The stretch factor applies horizontally, compressing the graph rather than stretching it vertically as it might initially appear.

Why Other Options Are Incorrect

Let’s consider why other possible descriptions might be incorrect. For example, a description involving a reflection over the x-axis would be incorrect because the negative sign is associated with the x-term inside the square root, indicating a reflection over the y-axis instead. Similarly, a translation to the right would be a misinterpretation of the “+9” term, which actually signifies a shift to the left.

Common Misconceptions

One common mistake is to confuse horizontal and vertical transformations. The transformations inside the function (affecting x) impact the graph horizontally, while transformations outside the function (affecting y) impact the graph vertically. Another misconception is the direction of horizontal translations; adding a constant inside the function shifts the graph to the left, not the right.

Correct Description of the Transformation

Given our analysis, the correct description of the graph of $y=\sqrt{-4x-36}$ compared to the parent square root function is:

• Horizontal compression by a factor of $\frac{1}{4}$
• Reflected over the y-axis
• Translated 9 units to the left

This combination of transformations accurately represents the changes applied to the parent function to obtain the given graph. Understanding the order and effect of these transformations is essential for mastering function transformations in mathematics.

Visualizing the Transformations

To further solidify understanding, visualizing these transformations can be incredibly helpful. Imagine starting with the basic square root function, $y = \sqrt{x}$. First, reflect it over the y-axis, which flips the graph horizontally. Next, compress it horizontally, making it narrower. Finally, shift the entire graph 9 units to the left. The resulting graph is the transformed function $y = \sqrt{-4x-36}$.

Importance of Order

The order in which transformations are applied is often crucial. In this case, the horizontal compression and reflection over the y-axis can be considered interchangeable in order since they both affect the x-variable. However, the translation must be applied after these scaling and reflection transformations to ensure it is applied to the transformed graph, not the original. Understanding the proper sequence ensures accurate graphing and interpretation of function transformations.

Conclusion

In conclusion, accurately describing the transformation of $y=\sqrt{-4x-36}$ compared to the parent square root function involves a detailed understanding of horizontal compressions, reflections over the y-axis, and horizontal translations. The graph undergoes a horizontal compression by a factor of $\frac{1}{4}$, is reflected over the y-axis, and is translated 9 units to the left. By breaking down the equation and analyzing each component, we can effectively map the transformations and avoid common pitfalls in function analysis. This approach not only helps in solving specific problems but also builds a stronger conceptual understanding of how functions behave under various transformations.