Transformasi Kurva: Translasi Y = X² + 6x Dengan T = (-2, 5)

Hey guys! So, you're here because you're probably wrestling with a math problem involving curve transformations, specifically with a translation. Don't sweat it; we'll break down how to handle the transformation of the curve y = x² + 6x when it's translated by T = (-2, 5). This concept is super important in understanding how shapes move around on a graph. This isn't just about getting the right answer; it's about seeing how the different parts of an equation affect the visual representation of that equation. In this article, we'll walk through the process step-by-step, making sure you grasp the underlying principles. We're going to use concepts of coordinate transformation to get this answer. We'll be using a mix of basic algebra and a bit of conceptual thinking. So, let's dive in and make sure you're totally comfortable with this type of problem.

Memahami Konsep Translasi dalam Transformasi Geometri

Alright, let's get down to the basics. Before we jump into the equation, it's super important to understand what a translation actually is in the world of geometry. Think of it like this: a translation is simply a slide. You're taking a shape (in our case, a curve defined by an equation) and moving it to a new position without changing its size, shape, or orientation. It's just a straight-up shift, like moving a piece on a board game. The direction and distance of this slide are defined by a vector, and that's exactly what T = (-2, 5) represents. The vector tells us how far and in which direction we're going to slide our curve. The vector components in T are used to modify the x and y values in the given equation. This is a fundamental concept in coordinate geometry, so grasping this is key to doing well in these kinds of problems.

So, what does T = (-2, 5) tell us? The first number, -2, means we're going to shift the curve 2 units to the left along the x-axis (because negative on the x-axis means left). The second number, 5, means we're going to shift the curve 5 units up along the y-axis (positive y means upwards). This is crucial, guys: the direction and magnitude of the translation. Understanding this will make the whole transformation process make a lot more sense. This is how we'll change our original equation. By the end of this, you should be able to look at any translation vector and instantly know how a curve will move. You'll understand how simple algebraic adjustments can make such big changes in the appearance of a graph.

Before we start transforming, it's worth reviewing the standard form of a quadratic equation. This way, we'll have a clear understanding of the original shape before the translation. Remember, the general form is y = ax² + bx + c. The original equation is y = x² + 6x, and a = 1, b = 6, and c = 0. This gives us a good grasp of the curve. The knowledge of the original curve is helpful in understanding the transformation effect. Remember, this translation will shift every point on the curve, so every point will change position.

Langkah-langkah Transformasi: Mengaplikasikan Translasi

Now, let's get down to business and actually transform that curve! We know our translation vector is T = (-2, 5). This means every point (x, y) on our original curve will be transformed to a new point (x', y') that has been shifted. To figure out the new equation, we have to consider how x and y change. The x-coordinate will be shifted to the left by 2 units, which is written as x' = x - 2. This is because we need to undo the translation. The y-coordinate will be shifted up by 5 units, which is written as y' = y + 5. So, to get the original coordinates in terms of the new ones, we can rewrite these as: x = x' + 2 and y = y' - 5. This is where the magic happens; we're essentially finding the equation of the translated curve by relating the new coordinates to the original equation.

Now, all we gotta do is substitute these values back into our original equation, y = x² + 6x. Where we see y, we substitute y' - 5; where we see x, we substitute x' + 2. This gives us y' - 5 = (x' + 2)² + 6(x' + 2). Remember, our goal is to find the equation of the transformed curve in terms of x and y (without the primes), so we'll simplify and rearrange the equation. This substitution is the heart of the transformation process. It changes how the original equation behaves. We're getting closer to our final solution. Now, let's simplify!

Let's get rid of those parentheses and simplify. Expand (x' + 2)², which gives us x'² + 4x' + 4. Next, distribute the 6 to get 6x' + 12. Our equation now looks like this: y' - 5 = x'² + 4x' + 4 + 6x' + 12. Now, we combine like terms. This means combining the x' terms (4x' and 6x') and the constants (4 and 12). So 4x' + 6x' = 10x' and 4 + 12 = 16. Our equation is now: y' - 5 = x'² + 10x' + 16. And, almost at the final step, add 5 to both sides to isolate y'. This gives us y' = x'² + 10x' + 21. This is the equation of our transformed curve. This is the heart of the question!

Because the primes are just there to help us see the relationship, we can drop them. We can rewrite it using regular x and y as y = x² + 10x + 21. This, my friends, is the equation of the curve y = x² + 6x after being transformed by the translation T = (-2, 5). This whole process might seem like a handful at first, but with practice, it'll become second nature. You'll become a transformation ninja, able to take any curve and move it around the coordinate plane with ease. Remember, the key is understanding how the translation vector affects the x and y coordinates and then substituting them back into the original equation. You got this!

Merangkum dan Memahami Hasil

So, what does all of this mean in terms of our original curve? The original curve was a parabola opening upwards. After the translation, the transformed curve is still a parabola that's exactly the same shape, just shifted. The vertex of the parabola, its lowest point, has also moved. If you were to graph both the original and transformed curves, you'd see the parabola has been shifted 2 units to the left and 5 units up, exactly as specified by the translation vector T = (-2, 5). Visualizing this is super important. This helps you confirm that your answer makes sense. When we shift the original curve y = x² + 6x, the new vertex has moved from (-3, -9) to (-5, -4). The shape is intact.

To solidify your understanding, try some practice problems. Change the translation vector and see how the curve changes. For example, what if T = (3, -1)? Or, try transforming a different type of equation, like a line or a circle. Also, practice questions that involve a reflection to gain a better overall understanding. Also, try doing it the other way around: start with the transformed equation, and figure out the original equation and the translation vector. This is a crucial skill. The better you get at these types of transformations, the better you'll become in geometry and calculus. Remember, math is all about understanding the concepts and seeing the relationships between them. These types of questions lay the foundation for a lot of advanced math topics. Keep practicing and keep asking questions, and you'll be a transformation master in no time! Remember to always check your answers graphically to make sure they make sense. Keep up the great work, and good luck with your math adventures!

Kesimpulan

So, we’ve successfully navigated the transformation of a curve through translation! We started with y = x² + 6x, applied the translation T = (-2, 5), and arrived at the transformed equation y = x² + 10x + 21. Remember, the key is understanding how the translation changes the x and y values, substituting those changes into the original equation, and simplifying. This process isn't just about getting the right answer; it's about building a solid understanding of how mathematical concepts work, and how they connect. Keep practicing, keep exploring, and you'll become a transformation pro in no time! You've successfully conquered the transformation challenge, guys! Keep up the awesome work!