Transformasi Fungsi: Dilatasi & Translasi Step-by-Step

Hey guys! Let's dive into the world of transformations, specifically how they affect a function. We're going to figure out the image of the function y = 3x + 2 when it's subjected to a dilation along the y-axis with a scale factor of 1/2, followed by a translation by (2, -3). Sounds complicated? Don't sweat it; we'll break it down into easy-to-understand steps. This is a common problem in mathematics, especially in coordinate geometry, so grasping this will be super helpful for your understanding.

Step 1: Understanding Dilation

Okay, first things first: dilation. What exactly does it do? Basically, dilation changes the size of a shape (or in our case, a function) while keeping its basic form. Think of it like zooming in or out on a picture. The scale factor determines how much the shape is enlarged or shrunk. In our problem, we have a dilation along the y-axis with a scale factor of 1/2. This means every y-coordinate of the function will be multiplied by 1/2. The x-coordinates remain unchanged during this specific type of dilation.

Now, let’s get into the nitty-gritty. If we have a point (x, y) on the original function y = 3x + 2, after the dilation, the x-coordinate stays as x, but the y-coordinate becomes (1/2)y. This is the core concept. We're compressing the function vertically. Imagine squishing the graph downwards. This gives us a new function, which we need to find. To do this, we need to express the original function in terms of the new coordinates. Since the new y-coordinate is (1/2)y, the original y is actually 2y', where y' is the new y-coordinate. We substitute this into our original function equation. Here's how we do it:

Original equation: y = 3x + 2 After dilation: We replace y with 2y'. So, 2y' = 3x + 2.

Now, to make it look like a standard function form, let's solve for y': y' = (3/2)x + 1. This is the equation after the dilation. See? Not so bad, right? We've successfully transformed our original function through dilation. This new equation, y' = (3/2)x + 1, describes the function after it's been scaled by a factor of 1/2 along the y-axis. The line is now 'closer' to the x-axis, which is what we'd expect.

Remember, understanding dilation is all about grasping how the scale factor affects each coordinate. For y-axis dilation, only the y-coordinates are directly affected, and the formula reflects this change. This step-by-step approach simplifies the process, making it less intimidating and more intuitive. Keep this basic understanding in mind as we transition to the next stage, translation. We're building a foundation here, and each concept builds on the other!

Step 2: Applying the Translation

Alright, now that we've dilated the function, it's time to translate it. Translation, in simpler terms, means sliding the function across the coordinate plane without changing its shape or size. Think of it as moving the entire graph horizontally and vertically. In our case, we're translating the function by the vector (2, -3). This means every point on the function will shift 2 units to the right and 3 units down. Easy, right?

So, with our new function from the dilation (y' = (3/2)x + 1), we have to incorporate the translation. If we start with a point (x', y') on the dilated function, the translation rule tells us that the new point (x'', y'') will be: x'' = x' + 2 (because we move 2 units to the right) and y'' = y' - 3 (because we move 3 units down). This is how the coordinates transform when we apply the translation.

Now, let's substitute. We already have y' = (3/2)x + 1. We know that x' = x'' - 2 (rearranging x'' = x' + 2). Substituting this into our dilated equation, we get: y' = (3/2)(x'' - 2) + 1.

Remember, we also have y'' = y' - 3, so y' = y'' + 3. Let's substitute that too: y'' + 3 = (3/2)(x'' - 2) + 1.

Now, we just need to simplify and solve for y''. Let's expand the brackets: y'' + 3 = (3/2)x'' - 3 + 1 y'' + 3 = (3/2)x'' - 2 y'' = (3/2)x'' - 5.

There you have it! The final equation after both the dilation and the translation is y'' = (3/2)x'' - 5. We often drop the double primes and just write y = (3/2)x - 5. This represents the transformed function, the final answer to our problem. This new equation shows the complete transformation – the combined effect of shrinking the graph along the y-axis and then shifting it to the right and down. The final function's line has a different y-intercept, which reflects both the dilation and the translation.

It is important to remember that the order in which these transformations occur matters. We first performed dilation, which scaled the function, and then translation, which shifted it. Both of these affect the final position of the line in the coordinate system, and we accounted for that in our calculations.

Step 3: Summarizing the Transformations

Let’s summarize what we’ve done and why it works. First, we started with a linear function, y = 3x + 2. This is our starting point. We then applied a dilation along the y-axis with a scale factor of 1/2. This resulted in a new function where the y-values were halved. The equation changed from y = 3x + 2 to y' = (3/2)x + 1. This initial transformation compressed the function vertically.

Next, we translated this dilated function by the vector (2, -3). This meant every point on the graph was shifted 2 units to the right and 3 units down. This transformation affects both the x and y coordinates. The translation process changes the y-intercept, which is why the equation finally became y = (3/2)x - 5. The final result is the function y = (3/2)x - 5. It's a testament to how multiple transformations change a function. Each transformation changes the function, and understanding how they interact is key.

In essence, we first 'squished' the graph downwards, then we 'slid' it to the right and downwards. Each step altered the function’s position in the coordinate plane. The process might seem long, but it highlights an essential skill in mathematics: applying mathematical concepts step-by-step. Break down complex problems, and you'll find they are not so difficult after all. Remember to always consider the order of transformations, as it can change the outcome. Also, try drawing the graphs before and after the transformation – it is always a good idea to visualize what's happening. Graphing the original function, the dilated function, and the translated function will reinforce your understanding of the concepts.

In a nutshell, we’ve successfully performed a combined transformation! We started with a simple linear function and, through dilation and translation, found its new form. This is a common type of problem in algebra and calculus and understanding it will give you a great advantage. This approach—understanding each step individually, and then combining them—is a great approach for a wide variety of mathematical problems. Keep practicing, and you'll become a transformation master in no time!