Transformations Of Functions A Detailed Analysis Of F(x)=2√(x+5)-3

In the realm of mathematical functions, transformations play a pivotal role in altering the shape and position of a graph. Understanding these transformations is crucial for analyzing and manipulating functions effectively. In this comprehensive guide, we will delve into the transformations applied to the function $f(x)=2 \sqrt{x+5}-3$, dissecting each component and its corresponding effect on the original square root function. We will explore the concepts of vertical stretch, horizontal translation, and vertical translation, providing a clear and concise explanation of how these transformations combine to produce the final graph. Whether you're a student grappling with function transformations or a seasoned mathematician seeking a refresher, this guide will equip you with the knowledge and insights to confidently navigate the world of function transformations.

Dissecting the Function: Unveiling the Transformations

To decipher the transformations applied to the function $f(x)=2 \sqrt{x+5}-3$, let's break it down step by step, comparing it to the parent function, $g(x) = \sqrt{x}$. The parent function serves as our baseline, the original square root function from which the transformations deviate. By comparing the transformed function to the parent function, we can isolate and identify the specific transformations that have been applied.

Vertical Stretch: Amplifying the Function

The first transformation we encounter is the vertical stretch by a factor of 2. This is represented by the coefficient '2' multiplying the square root term. A vertical stretch essentially amplifies the y-values of the function, making the graph appear taller. In simpler terms, imagine stretching the graph vertically, away from the x-axis. Each y-coordinate is multiplied by the stretch factor, in this case, 2. For example, if a point on the parent function has coordinates (x, y), the corresponding point on the transformed function after the vertical stretch will be (x, 2y). This transformation doesn't affect the x-values, only the vertical dimension of the graph.

Horizontal Translation: Shifting the Function Left

Next, we encounter the horizontal translation, represented by the '+5' inside the square root. This might seem counterintuitive, but a '+5' inside the function actually shifts the graph 5 units to the left. This is because the transformation affects the x-values before the square root is applied. To understand why it shifts left, consider the value of x that makes the expression inside the square root zero. For the parent function, $g(x) = \sqrt{x}$, the square root is zero when x = 0. However, for $f(x)=2 \sqrt{x+5}-3$, the square root is zero when x + 5 = 0, which means x = -5. This indicates that the graph has been shifted 5 units to the left, so that the starting point of the square root function is now at x = -5 instead of x = 0.

Vertical Translation: Shifting the Function Down

Finally, we have the vertical translation represented by the '-3' outside the square root. This transformation shifts the entire graph 3 units down. This is a more straightforward transformation; the constant term added or subtracted outside the function directly shifts the graph vertically. A negative constant shifts the graph down, while a positive constant shifts it up. In our case, subtracting 3 from the entire function means that every y-value on the graph is decreased by 3, resulting in a downward shift of the graph by 3 units.

Putting It All Together: The Combined Transformations

Now that we've dissected each individual transformation, let's combine them to understand the overall effect on the graph of the function. The function $f(x)=2 \sqrt{x+5}-3$ is obtained from the parent function $g(x) = \sqrt{x}$ by applying the following transformations in sequence:

1. Horizontal Translation: Shift the graph 5 units to the left.
2. Vertical Stretch: Stretch the graph vertically by a factor of 2.
3. Vertical Translation: Shift the graph 3 units down.

The order in which these transformations are applied is important. Horizontal and vertical stretches/compressions should generally be performed before translations. This is because stretches and compressions affect the scale of the graph, while translations simply shift its position. By applying the stretch before the translation, we ensure that the scaling is applied to the translated graph, rather than the other way around.

Identifying the Correct Transformation Description

Now, let's revisit the original question and the given options. The question asks us to identify the correct description of the transformations applied to $f(x)=2 \sqrt{x+5}-3$. Based on our analysis, the correct description is:

• Vertical stretch by a factor of 2, horizontal translation left 5, vertical translation down 3

Let's examine why the other options are incorrect:

• Option B: Vertical stretch by a factor of 2, horizontal translation right 5, vertical translation down 3

  This option is incorrect because it states that the horizontal translation is to the right by 5 units. As we discussed earlier, the '+5' inside the square root results in a translation to the left by 5 units.

• Option C: Vertical compression

  This option is incorrect because the function undergoes a vertical stretch, not a vertical compression. The coefficient '2' multiplying the square root term indicates a stretch, not a compression. A vertical compression would be represented by a coefficient between 0 and 1.

• Option A: Correct transformation not listed

  This option would be correct only if none of the other options accurately described the transformations. However, we have already identified the correct description, so this option is incorrect.

Common Pitfalls and Key Takeaways

Understanding function transformations can be challenging, and there are some common pitfalls to avoid. One common mistake is confusing the direction of horizontal translations. Remember that a '+c' inside the function results in a shift to the left, while a '-c' results in a shift to the right. Another common mistake is applying the transformations in the wrong order. Stretches and compressions should generally be applied before translations.

Key takeaways from this guide include:

• A coefficient greater than 1 multiplying the function results in a vertical stretch.
• A coefficient between 0 and 1 multiplying the function results in a vertical compression.
• A '+c' inside the function results in a horizontal translation to the left by c units.
• A '-c' inside the function results in a horizontal translation to the right by c units.
• A '+c' outside the function results in a vertical translation up by c units.
• A '-c' outside the function results in a vertical translation down by c units.

By mastering these concepts and avoiding common pitfalls, you can confidently analyze and manipulate functions through transformations.

Conclusion: Mastering Function Transformations

In conclusion, the function $f(x)=2 \sqrt{x+5}-3$ undergoes a vertical stretch by a factor of 2, a horizontal translation 5 units to the left, and a vertical translation 3 units down. By carefully dissecting the function and understanding the individual effects of each transformation, we can accurately describe the overall transformation applied to the graph. Mastering function transformations is a fundamental skill in mathematics, enabling us to analyze and manipulate functions effectively. This guide has provided a comprehensive explanation of the transformations applied to the given function, equipping you with the knowledge and insights to confidently tackle similar problems in the future. Remember to practice applying these concepts to various functions to solidify your understanding and further enhance your mathematical prowess.

By understanding the principles outlined in this guide, you'll be well-equipped to decipher transformations of functions and analyze their graphical representations. Remember to consider the order of transformations, the impact of coefficients and constants, and the parent function as your reference point. With practice and a solid grasp of these concepts, you'll be able to confidently navigate the world of function transformations and unlock a deeper understanding of mathematical relationships.