Graph Transformation: Analyzing Y=√(−4x−36)

Hey everyone! Today, we're diving into the fascinating world of graph transformations, specifically focusing on how the function $y = \sqrt{-4x - 36}$ compares to its parent function, the simple square root function, $y = \sqrt{x}$. This might seem intimidating at first, but trust me, we'll break it down step-by-step and make it super clear. We're going to explore how different operations within the equation affect the graph's shape, orientation, and position on the coordinate plane. So, buckle up and get ready to transform your understanding of graph transformations! Let's get started and unravel this mathematical mystery together! We’ll dissect the equation, identify the transformations, and visualize how they play out on the graph. By the end of this, you'll be able to confidently describe the transformations applied to the square root function and understand the impact of each transformation. It's like becoming a graph transformation detective – spotting the clues and piecing together the puzzle.

Unveiling the Parent Function: The Foundation of Our Transformation Journey

Before we tackle the transformed function, let's solidify our understanding of the parent square root function, $y = \sqrt{x}$. This is the bedrock upon which our transformation understanding will be built. Think of it as the original blueprint before any modifications are made. The graph of $y = \sqrt{x}$ starts at the origin (0, 0) and gracefully curves upwards and to the right. It only exists for non-negative values of x because we can't take the square root of a negative number (in the realm of real numbers, at least!). So, the domain is $x \geq 0$, and the range is $y \geq 0$. This basic shape is crucial. We need to visualize this gentle curve because all the transformations we'll discuss are relative to this parent function. Understanding its starting point and direction will make it much easier to track the changes caused by the transformations. You might even want to sketch a quick graph of $y = \sqrt{x}$ on a piece of paper. Having that visual reference will make the following explanations click even more. We'll be referring back to this parent function repeatedly, so make sure you've got a good mental image of it.

Furthermore, it's helpful to remember a few key points on the parent function. For example, when x is 0, y is 0; when x is 1, y is 1; and when x is 4, y is 2. These points act as anchors, helping us see how the transformed graph stretches, shifts, or reflects compared to its original form. Now that we have a solid grasp of the parent function, we're ready to delve into the transformations lurking within the equation $y = \sqrt{-4x - 36}$. Get ready to put on your transformation goggles!

Decoding the Transformations: A Step-by-Step Analysis

Now comes the exciting part: dissecting the equation $y = \sqrt{-4x - 36}$ to uncover the transformations. To make things easier, let's rewrite the equation slightly by factoring out a -4 from inside the square root: $y = \sqrt{-4(x + 9)}$. This seemingly small change makes the transformations much clearer to identify. Remember, transformations happen in a specific order, and following that order will prevent confusion. We'll tackle them one at a time, describing the effect of each on the parent function. Think of it like peeling back the layers of an onion, revealing each transformation step-by-step.

The first transformation we encounter is the horizontal stretch/compression and reflection due to the -4 inside the square root. The -4 acts on the x-value before the square root is taken, so it affects the graph horizontally. The negative sign indicates a reflection over the y-axis. Instead of the graph opening to the right (as the parent function does), it will now open to the left. The coefficient 4 implies a horizontal compression by a factor of $\frac{1}{4}$. This means the graph is squeezed horizontally towards the y-axis. It's crucial to remember that horizontal transformations do the opposite of what you might intuitively expect. A number greater than 1 compresses the graph, while a number between 0 and 1 stretches it.

Next, we have the horizontal translation caused by the +9 inside the parentheses. Again, this affects the x-values, so it's a horizontal transformation. The +9 indicates a shift of 9 units to the left. Remember, horizontal translations also behave counterintuitively. A positive value inside the parentheses shifts the graph to the left, and a negative value shifts it to the right. This shift is crucial in understanding the final position of the graph. It determines where the graph begins its journey on the coordinate plane. So, combining these transformations, we see the graph is reflected over the y-axis, compressed horizontally by a factor of $\frac{1}{4}$, and shifted 9 units to the left. We're getting closer to fully describing the transformation!

Putting It All Together: Describing the Transformed Graph

Alright, let's synthesize what we've learned and describe the graph of $y = \sqrt{-4x - 36}$ compared to the parent square root function. We've identified three key transformations: a reflection over the y-axis, a horizontal compression by a factor of $\frac{1}{4}$, and a horizontal translation of 9 units to the left. It's important to state these transformations in the correct order to accurately describe the final graph. Think of it as telling a story – you need to get the sequence of events right for the story to make sense.

The graph of $y = \sqrt{-4x - 36}$ is the graph of $y = \sqrt{x}$ reflected over the y-axis, compressed horizontally by a factor of $\frac{1}{4}$, and translated 9 units to the left. Notice how we clearly stated each transformation and its effect on the parent function. This level of detail is crucial for a complete and accurate description. Guys, you can even visualize this in your mind: imagine the parent square root function flipping over the y-axis, then being squeezed horizontally, and finally sliding 9 units to the left. That mental picture should match the final transformed graph.

Now, let's take a look at the options provided and see which one accurately matches our description. Option A mentions a stretch by a factor of 2 and a reflection over the x-axis, which we didn't find in our analysis. Option B, however, sounds much more promising as it includes the correct types of transformations. By carefully comparing our analysis with the answer choices, we can confidently select the one that perfectly captures the transformations we've identified. Remember, the key is to break down the equation, identify each transformation, and describe its effect on the parent function in a clear and concise manner. You've got this!

Identifying the Correct Option: A Moment of Triumph

Now, let's apply our knowledge to the answer choices provided in the original question. After carefully analyzing the transformations, we arrived at the following description: the graph of $y = \sqrt{-4x - 36}$ is the graph of $y = \sqrt{x}$ reflected over the y-axis, compressed horizontally by a factor of $\frac{1}{4}$, and translated 9 units to the left. Let's examine the options to see which one aligns perfectly with our findings.

• Option A: stretched by a factor of 2, reflected over the x-axis, and translated 9 units right. This option includes incorrect transformations. We identified a horizontal compression, not a stretch, and the reflection is over the y-axis, not the x-axis. The translation is also in the wrong direction. So, Option A is not the correct answer.
• Option B: stretched by a factor of 2, reflected over the Discussion category:

Unfortunately, the provided text ends abruptly, so we cannot fully evaluate Option B. However, based on our analysis, we know the correct description involves a reflection over the y-axis, a horizontal compression by a factor of $\frac{1}{4}$, and a translation of 9 units to the left. To definitively choose the correct option, we would need the complete text of Option B and any subsequent options. But the process remains the same: compare each option to our detailed description of the transformations and select the one that matches perfectly. This methodical approach ensures you arrive at the correct answer with confidence. Great job working through this problem, guys! Keep practicing, and you'll become a graph transformation master in no time!