Transforming Linear Equations: Step-by-Step Guide

Hey guys, let's break down how we get from the graph of y = -3x - 4 to y = -3/2x - 4. It's all about understanding transformations, and we'll make sure it's super clear. No sweat, we'll get through it together! The original question is actually asking about how the graph of a line changes when you tweak its equation. The key is to identify the transformations, which are shifts, stretches, compressions, and reflections, that turn the original line into the new one. In this case, we're dealing with changes to the slope and how that impacts the horizontal dimensions of the graph. So, let's figure out what’s happening with the x-values, that’s where the magic happens! Remember, we're looking at how the graph changes, not just solving for x or y.

Understanding the Original Equation: y = -3x - 4

First off, let's get friendly with the original equation: y = -3x - 4. This is a linear equation, and it represents a straight line. In this equation, -3 is the slope and -4 is the y-intercept. The slope tells us how steeply the line rises or falls. A slope of -3 means that for every 1 unit we move to the right on the x-axis, we go down 3 units on the y-axis. The y-intercept, -4, tells us where the line crosses the y-axis—at the point (0, -4). It's always a great idea to visualize the line or, better yet, sketch it to have a visual reference. Understanding these fundamentals is super important. This is our starting point, our baseline. Now, let's think about how that baseline changes with different transformations. We're going to focus on how the slope changes the horizontal dimensions of the graph.

Visualizing the Initial Graph

Imagine this line. It goes down pretty steeply, right? Starting at -4 on the y-axis, it drops sharply as we move to the right. That steepness comes from the slope of -3. This will help us understand the impact of the transformation.

Analyzing the Transformed Equation: y = -3/2x - 4

Now, let's look at the transformed equation: y = -3/2x - 4. Notice that the y-intercept is still -4. That means the line still crosses the y-axis at the same point. The only thing that has changed is the slope. The slope is now -3/2. This is less steep than the original slope of -3. A slope of -3/2 means that for every 2 units we move to the right, we go down 3 units. This indicates a change in the steepness of the line. Since the y-intercept is the same, the entire line will rotate or change steepness around that point, making the line appear to stretch or compress horizontally.

Comparing Slopes and Impact

The main difference is in the slope. The original slope was -3, and the new slope is -3/2. Since the denominator of the new slope is smaller, and the absolute value of the slope is smaller than the original slope, the line is less steep. The absolute value of the new slope (3/2) is less than the absolute value of the original slope (3), meaning the line is less steep. In terms of transformations, this indicates a horizontal stretching or compression.

Deciphering the Transformation

So, how do we get from y = -3x - 4 to y = -3/2x - 4? The correct answer involves a horizontal stretch or compression. Think about it this way: the y-intercept stays the same, but the line becomes less steep. That means it is horizontally stretched or compressed. In this case, the line is less steep, suggesting a horizontal stretch. But is it a stretch or a compression? Let's figure it out!

Understanding Horizontal Stretching and Compression

Horizontal stretching makes the graph wider, while horizontal compression makes it narrower. Since the slope changed, we know that the line has been altered to a less steep inclination, which is a visual aspect that could result in a horizontal stretch. The fact that the y-intercept remains unchanged indicates that we're focusing on changes parallel to the x-axis.

Detailed Breakdown of the Transformation

Let's look at the slope change more closely. We're going from a slope of -3 to -3/2. Here’s where it gets a bit tricky, so pay attention! The general form for horizontal stretching or compression involves changing the x coefficient in the equation. The change in the slope of the line indicates a horizontal stretch. The question is, by what factor? The original slope is -3, and the new slope is -3/2. The ratio of the new slope to the original slope gives us the factor of stretch or compression. In our case, (-3/2) / (-3) = 1/2. This means that the graph is stretched horizontally by a factor of 2. Since the question is about a transformation, this is the correct answer. The graph is stretched by a factor of 2.

Evaluating the Answer Choices

Let’s look at the answer choices to see which one fits the bill. A. The graph is stretched horizontally by a factor of 2 and then moved right 4 units. This one sounds promising, as we have a horizontal stretch. Let's analyze the other options.

B. The graph is compressed horizontally by a factor of 2 and then moved down 4 units. This suggests compression and a vertical shift, which don’t align with our analysis.

Considering our analysis, the correct answer must involve a horizontal stretch. Let's analyze this more directly.

• Option A: The graph is stretched horizontally by a factor of 2 and then moved right 4 units. This aligns with the changes in slope, and is a possible answer. The horizontal stretch is the main focus here.

Confirming the Correct Answer

So, based on our calculations, the graph is indeed stretched horizontally by a factor of 2. Therefore, the graph of y=-3x-4 is transformed to produce the graph of y = -3/2x-4 by stretching horizontally by a factor of 2. The correct answer must be associated with the horizontal stretch.

The Correct Answer Explained

• A. The graph is stretched horizontally by a factor of 2 and then moved right 4 units. This is not the entire answer. It is only a part of the transformation. Moving the graph right 4 units is incorrect.

Conclusion

Alright, guys, we’ve cracked it! The correct answer involves a horizontal stretch. We figured out how the change in the slope affects the graph's shape. We have a clear understanding now!

Hopefully, this explanation helped you understand how these transformations work. Keep practicing, and you'll become a pro at these types of problems in no time!