Graph Transformation What Converts F(x)=-4x+8 To G(x)=-2x+8

Hey guys! Today, we're diving into a fun little math problem that involves graph transformations. Specifically, we're trying to figure out what kind of transformation takes the graph of the function f(x) = -4x + 8 and turns it into the graph of g(x) = -2x + 8. We've got four options to choose from: a vertical shrink, a horizontal stretch, a horizontal shrink, or a vertical stretch. Let's break it down and see what's really going on here. To solve this, let’s analyze the two linear functions, f(x) = -4x + 8 and g(x) = -2x + 8. Notice anything similar? Yep, they both have the same y-intercept, which is 8. That means both lines cross the y-axis at the same point. What’s different then? The slopes! The slope of f(x) is -4, and the slope of g(x) is -2. This is where the transformation magic happens. Remember that the slope affects the steepness of the line. A steeper line has a larger absolute value of the slope. So, f(x) is steeper than g(x) because |-4| is greater than |-2|. This means that to get from f(x) to g(x), we need to make the line less steep. Think of it like squishing the graph towards the x-axis. This kind of transformation can be a bit tricky to visualize, so let's explore the options.

Understanding Vertical and Horizontal Transformations

Before we jump to the answer, let’s quickly recap the difference between vertical and horizontal transformations. This is crucial for understanding how functions behave and transform. It might sound a bit technical, but trust me, it's super useful stuff! Let’s start with vertical transformations. These are the ones that affect the y-values of the function. Think of it as stretching or compressing the graph up and down. A vertical stretch makes the graph taller, while a vertical shrink makes it shorter. Imagine you're grabbing the graph from the top and bottom and either pulling it apart (stretch) or pushing it together (shrink). The key thing here is that the x-values stay the same, but the y-values change. For example, if you have a function y = f(x) and you multiply the entire function by a constant greater than 1 (like 2), you're doing a vertical stretch. If you multiply by a constant between 0 and 1 (like 0.5), you're doing a vertical shrink. Now, let’s talk about horizontal transformations. These are the ones that affect the x-values. Think of it as stretching or compressing the graph left to right. A horizontal stretch makes the graph wider, while a horizontal shrink makes it narrower. It’s like grabbing the graph from the sides and either pulling it apart (stretch) or pushing it together (shrink). The funny thing about horizontal transformations is that they often feel counterintuitive. When you do something to the x inside the function, it affects the graph in the opposite way you might expect. For example, if you have a function y = f(x) and you replace x with 2x, you're actually doing a horizontal shrink by a factor of 1/2. If you replace x with (1/2)x, you're doing a horizontal stretch by a factor of 2. Why the opposite effect? It’s because you're changing the input values needed to get the same output. With these basics in mind, let's get back to our original problem. We need to figure out what kind of transformation changes the slope of the line from -4 to -2 while keeping the y-intercept the same. Which type of transformation do you think is the culprit?

Analyzing the Options: Which Transformation Fits?

Let's dive into each option and see which one aligns with the change from f(x) = -4x + 8 to g(x) = -2x + 8. Remember, our goal is to make the line less steep. We’re essentially going from a steeper negative slope (-4) to a less steep negative slope (-2). This means we need to figure out if this change is due to squishing the graph vertically or horizontally. First up, let’s consider Option A: Vertical Shrink. A vertical shrink compresses the graph towards the x-axis. This means the y-values get closer to zero. If we vertically shrink f(x), we're essentially making the line less tall, but would it directly change the slope in the way we need? Think about it. If you squish a line vertically, you're changing the rise over the run. While it might make the line look less steep in some ways, the direct mathematical effect on the slope isn’t as straightforward as other transformations. So, while a vertical shrink could play a role, it might not be the most direct answer here. Now let’s examine Option B: Horizontal Stretch. A horizontal stretch pulls the graph wider away from the y-axis. Remember, horizontal transformations can be a bit tricky. Stretching horizontally actually compresses the x-values, and vice versa. So, if we stretch f(x) horizontally, we are, in effect, making the line less steep. This is because the same change in y now happens over a larger change in x. This is definitely a contender! Let’s hold onto this thought and move to the next option. Next is Option C: Horizontal Shrink. A horizontal shrink pushes the graph closer to the y-axis, compressing it from the sides. This makes the line steeper, which is the opposite of what we want. We're trying to make the line less steep, not more. So, a horizontal shrink is not the right transformation for our problem. It would actually take g(x) and turn it into something closer to f(x). Finally, let’s look at Option D: Vertical Stretch. A vertical stretch pulls the graph taller away from the x-axis. This makes the line steeper, just like a horizontal shrink. Again, this is not what we need. We need to decrease the steepness, not increase it. A vertical stretch would move us further away from g(x). So, after analyzing all the options, which one seems the most likely to transform f(x) into g(x)?

The Correct Transformation: Horizontal Stretch

Alright, after carefully examining all the options, the transformation that converts the graph of f(x) = -4x + 8 into the graph of g(x) = -2x + 8 is a B. horizontal stretch. Let's break down why this is the case and solidify our understanding. Remember how we talked about horizontal transformations being a bit counterintuitive? This is a perfect example. When we stretch a graph horizontally, we're essentially pulling it wider. In terms of the equation, this means we're affecting the x-values. To transform f(x) into g(x), we need to effectively halve the slope. Think of it this way: if we stretch the graph horizontally by a factor of 2, it's like we're making the x-axis twice as long. This means that for the same change in y, we need twice the change in x. This directly affects the slope, making it half of what it was originally. Mathematically, if we replace x in f(x) with (1/2)x, we get: f((1/2)x) = -4(1/2)x + 8 = -2x + 8 = g(x). See how that works? By stretching the graph horizontally, we've successfully transformed f(x) into g(x). Now, why not the other options? A vertical shrink would make the graph less tall, but it doesn't directly change the slope in the same way a horizontal stretch does. A horizontal shrink would make the graph steeper, which is the opposite of what we want. And a vertical stretch would also make the graph steeper. So, a horizontal stretch is the perfect fit for this transformation. It’s all about understanding how changes in x and y affect the slope and the overall shape of the graph. So, the next time you encounter a similar problem, remember to think about how the graph is being stretched or compressed, and whether that's happening horizontally or vertically. You've got this!

Final Thoughts and Key Takeaways

So, to wrap things up, we've successfully identified that a horizontal stretch is the transformation that converts the graph of f(x) = -4x + 8 into the graph of g(x) = -2x + 8. This problem highlights a really important concept in math: understanding how transformations affect functions. It's not just about memorizing rules, but about visualizing what's happening to the graph. Think about how the x and y values are changing, and how that impacts the shape and steepness of the function. We covered a lot of ground today, from understanding vertical and horizontal transformations to analyzing different options and applying them to a specific problem. Remember, the key takeaways are: Vertical transformations affect the y-values (up and down). Horizontal transformations affect the x-values (left and right). Horizontal transformations often behave in a counterintuitive way. Stretching compresses values, and shrinking stretches them. The slope of a line is directly affected by horizontal stretches and shrinks. By understanding these concepts, you'll be well-equipped to tackle a wide range of transformation problems. And remember, math is like a puzzle. It's all about breaking down the problem, analyzing the pieces, and putting them together to find the solution. Keep practicing, keep exploring, and most importantly, keep having fun with it! You guys are doing awesome. Keep up the great work, and I'll catch you in the next math adventure!