Transformations Of Functions: Translation And Dilation

Hey guys! Let's dive into a cool math problem that combines translation and dilation transformations. We'll start with a function and see how it changes when we shift it and stretch it. This is super useful in understanding how graphs behave, so let's get to it!

Understanding the Original Function

First, let’s talk about the original function we’re starting with: f(x) = 2x² + 1. This is a quadratic function, which means its graph is a parabola. The 2x² part tells us that the parabola opens upwards, and the + 1 shifts the whole graph up by one unit on the y-axis. So, the vertex (the lowest point) of this parabola is at (0, 1). Understanding this original function is key because it’s what we’re going to transform. We need to know where it starts to understand how the transformations change it. Think of it like a base recipe; you need to know the original recipe before you can start adding and changing ingredients to create something new and exciting! The coefficient '2' in front of x² also affects how narrow or wide the parabola is. A larger coefficient makes the parabola narrower, while a smaller one makes it wider. This is because the y values increase (or decrease if the coefficient is negative) more rapidly as x moves away from zero. So, in our case, the parabola is a bit narrower than the standard x² parabola. This initial understanding helps us visualize what the graph looks like and how the transformations will affect its shape and position. Remember, transformations are all about moving and changing the shape of the original graph, so knowing the original shape is half the battle!

Translation Transformation

Alright, so the first transformation we're doing is a translation. Specifically, we're translating by the vector ${\begin{pmatrix}-4\\ 0\end{pmatrix}}$. What does this mean, exactly? Well, a translation just means we're moving the entire graph without rotating or resizing it. The vector tells us how much to move it horizontally and vertically. In this case, (-4, 0) means we're shifting the graph 4 units to the left (because of the -4) and not moving it up or down at all (because of the 0). So, every point on the original graph y = f(x) is going to move 4 units to the left. To represent this mathematically, we replace x in the original function with (x + 4). Why (x + 4) instead of (x - 4)? Because we're shifting to the left, we need to increase the x value to get to the same y value as before. So, our new function after the translation is y = f(x + 4) = 2(x + 4)² + 1. This new equation represents the translated graph. If you were to plot this, you’d see the exact same parabola as before, just shifted 4 units to the left. The vertex, which was at (0, 1), is now at (-4, 1). Understanding translations is crucial because it's a fundamental transformation. It's like picking up a drawing and moving it on a piece of paper; the drawing itself doesn't change, just its location. This concept is used extensively in computer graphics, physics simulations, and many other fields. So, mastering translations gives you a solid foundation for more advanced transformations.

Dilation Transformation

Next up, we have dilation. In this problem, we're dilating parallel to the y-axis with a scale factor of 3. What this means is that we're stretching the graph vertically by a factor of 3. Imagine grabbing the graph from the top and bottom and pulling it apart, making it taller. So, how do we represent this mathematically? To dilate parallel to the y-axis, we multiply the entire function by the scale factor. In this case, we multiply the translated function y = 2(x + 4)² + 1 by 3. This gives us our final transformed function: y = 3[2(x + 4)² + 1] = 6(x + 4)² + 3. Notice that the y value of every point on the graph is now three times its previous value. For example, the vertex of the translated parabola was at (-4, 1). After the dilation, the vertex is now at (-4, 3). The entire parabola has been stretched vertically, making it taller and narrower compared to the translated graph. Dilation is another key transformation that changes the size of the graph. A scale factor greater than 1 stretches the graph, while a scale factor between 0 and 1 compresses it. Understanding dilation is important in various applications, such as image processing, where you might want to zoom in or out on an image. It also plays a role in understanding how quantities scale in different physical systems. So, mastering dilation helps you understand how transformations affect the size and shape of graphs.

Combining Transformations

Okay, let's recap what we've done so far. We started with the function f(x) = 2x² + 1. First, we translated it by the vector ${\begin{pmatrix}-4\\ 0\end{pmatrix}}$, which shifted the graph 4 units to the left, resulting in the function y = 2(x + 4)² + 1. Then, we dilated the translated graph parallel to the y-axis with a scale factor of 3, which stretched the graph vertically, giving us the final transformed function y = 6(x + 4)² + 3. So, the key to solving this problem was understanding the order of operations and how each transformation affects the function. Translation shifts the graph, while dilation stretches or compresses it. By applying these transformations step-by-step, we were able to find the final transformed function. This combined transformation is a powerful tool for manipulating graphs and understanding how different functions relate to each other. In many real-world applications, transformations are used to model changes in data, optimize designs, and analyze complex systems. For example, in computer graphics, transformations are used to rotate, scale, and position objects in 3D space. In signal processing, transformations are used to analyze and manipulate signals, such as audio and video. So, understanding combined transformations is a valuable skill that can be applied in a wide range of fields.

Final Transformed Function

So, after applying the translation and dilation, the final transformed function is y = 6(x + 4)² + 3. This equation represents the parabola after it has been shifted 4 units to the left and stretched vertically by a factor of 3. You can graph this function to visualize the changes. The vertex of the transformed parabola is at (-4, 3), and the parabola is narrower than the original. And that's it! We've successfully transformed the original function using a combination of translation and dilation. Remember, the order of transformations matters, so make sure to apply them in the correct sequence. Keep practicing, and you'll become a transformation master in no time!

Visualizing the Transformations

To really nail this down, let's talk about visualizing these transformations. Imagine the original parabola, f(x) = 2x² + 1. It's sitting there, symmetrical around the y-axis, with its lowest point at (0, 1). Now, picture grabbing that parabola and sliding it 4 units to the left. That's the translation! The whole shape moves, but it doesn't change size or orientation. Now, imagine grabbing the translated parabola from the top and bottom and stretching it upwards, like pulling taffy. That's the dilation! The parabola gets taller and skinnier. Visualizing these transformations can make them much easier to understand. You can even use graphing software to plot the original function, the translated function, and the final transformed function. This will allow you to see the changes in real-time and reinforce your understanding of how each transformation affects the graph. Experiment with different transformations and different functions to see how they interact. The more you visualize and experiment, the more intuitive these concepts will become.

Importance of Order

One thing that's super important to remember is that the order of transformations matters! If we had dilated first and then translated, we would have ended up with a different final function. Transformations don't always commute, which means transformation A followed by transformation B is not always the same as transformation B followed by transformation A. In our case, we translated first and then dilated. If we had dilated first, we would have multiplied the original function by 3, getting y = 3(2x² + 1) = 6x² + 3. Then, we would have translated this function 4 units to the left, replacing x with (x + 4), resulting in y = 6(x + 4)² + 3. In this specific case, it seems like we got the same answer. However, this is not always the case, especially when dealing with more complex transformations or different types of functions. To avoid errors, always follow the order of transformations specified in the problem. If the order is not specified, it's best to try both orders and see if they give the same result. If they don't, then the order matters, and you need to pay close attention to the problem statement. Understanding the importance of order is crucial for mastering transformations and applying them correctly in various applications.

Practice Makes Perfect

Transformations of functions might seem a bit abstract at first, but with practice, they become much easier to grasp. Try working through more examples with different types of functions and different transformations. Experiment with translations, dilations, reflections, and rotations. See how each transformation affects the graph and the equation of the function. Use graphing software to visualize the transformations and check your answers. The more you practice, the more comfortable you'll become with these concepts. You can also try creating your own transformation problems and solving them. This will help you develop a deeper understanding of the underlying principles and improve your problem-solving skills. Remember, math is not a spectator sport; you need to actively participate to truly learn it. So, grab a pencil and paper, start working through some examples, and have fun exploring the world of transformations! And that's a wrap, folks! Keep up the great work, and I'll see you in the next problem!