Graph Transformation: Shifting Parabolas Explained

Hey math enthusiasts! Let's dive into a cool concept: graph transformations. Specifically, we're going to break down how to move the graph of a function around on the coordinate plane. This is super handy for understanding how changes in an equation affect its visual representation. We'll use the example of parabolas, those U-shaped curves, to make it crystal clear. So, get ready to shift your perspective – literally!

Understanding the Basics: Parent Functions and Transformations

Alright, before we jump into the nitty-gritty, let's set the stage. Every function has a "parent function." This is the simplest form of the function. For a quadratic function like the one we're dealing with, f(x) = 2x² + 4, the parent function is f(x) = x². Think of it as the foundation. Now, transformations are what we do to that parent function – we stretch it, shrink it, flip it, or, as in our case, slide it around. These transformations are determined by the numbers and operations in the equation. There are a couple of main types of transformations: translations, which are just shifts left, right, up, or down; reflections, which flip the graph across an axis; and dilations, which stretch or compress the graph. Today, we'll focus on translations.

Now, let's look at the given problem. We are given two functions, $f(x) = 2x^2 + 4$ and $g(x) = 2(x-9)^2 + 4$. The question is, what kind of transformation converts the graph of $f(x)$ into the graph of $g(x)$? Let's start by looking at the equation of f(x). Here, we can see the coefficient '2' in front of the $x^2$ term, this impacts the vertical stretch or compression and a vertical shift of 4 units up. The function g(x) is slightly different. Here, we also have the coefficient '2' in front of the $(x-9)^2$ term, which means that the function will have the same vertical stretch as the function f(x). However, inside the parentheses, we see (x-9). This indicates a horizontal shift. Remember, when you see a change directly to the x inside the function, like (x-9), it impacts the graph horizontally. Since it's x-9, it's going to shift the graph 9 units to the right. Keep in mind that the horizontal shifts are a bit counterintuitive, so always make sure you’re interpreting them correctly. Also, the '+ 4' at the end still tells us that the entire graph is shifted up 4 units.

So, how does the function change? The presence of (x-9) within the square in g(x) is a horizontal translation. The other parts of the equation haven't changed the graph, meaning the vertical stretch and vertical shift are the same. Basically, the transformation takes the original parabola and slides it horizontally. We could have also used the vertex form of a parabola, y = a(x-h)^2 + k. Here, the vertex of the parabola is at the point (h, k). This shows us the horizontal and vertical shifts. This tells us that the vertex of f(x) is at (0, 4), and the vertex of g(x) is at (9, 4). Now, do you see it? g(x) is simply the graph of f(x) moved 9 units to the right. Got it?

Decoding the Options: Left, Right, Up, or Down?

Now, let's match our understanding with the multiple-choice options. The goal is to figure out how the graph of f(x) is transformed to get the graph of g(x). Each option describes a type of translation. Let's break them down one by one:

A. Translation 9 units left: This means shifting the entire graph of f(x) to the left. If we were to shift f(x) to the left, the equation would change to something like g(x) = 2(x+9)² + 4. Notice the plus sign inside the parentheses. This option isn't what we're looking for.

B. Translation 9 units down: This involves moving the entire graph downwards. This would change the equation in a way that's not compatible with our problem. For example, the equation would be g(x) = 2x² + 4 - 9. This isn't the transformation we need to get from f(x) to g(x).

C. Translation 9 units up: This option suggests moving the entire graph upwards. Similar to option B, this isn't what we are looking for. This would involve adding 9 to the function f(x), like g(x) = 2x² + 4 + 9. But we need to shift it horizontally.

D. Translation 9 units right: Bingo! This is the one. This option aligns perfectly with our analysis. As we mentioned, the (x-9) inside the squared term tells us the graph is shifted 9 units to the right. This is the same transformation that would turn f(x) into g(x). So, we can see that the correct answer is D.

Putting it All Together: The Big Picture

So, what did we learn today, guys? We figured out that to get from the graph of f(x) = 2x² + 4 to the graph of g(x) = 2(x-9)² + 4, we need to perform a translation of 9 units to the right. Understanding how these small changes in an equation affect the overall shape and position of a graph is super important. It's the key to unlocking the secrets of functions and their graphical representations. Keep practicing, and you'll become a graph transformation pro in no time! Remember, when you see something like (x - h), it means a horizontal shift. If it's x - 9, then you shift to the right by 9 units. Keep in mind the difference between horizontal and vertical shifts, and you'll be able to tackle many more problems like this one. Stay curious, and keep exploring the amazing world of math!