Understanding Quadratic Transformations: F(x) To G(x)

Hey everyone! Let's dive into the awesome world of quadratic functions and figure out how to transform them. We're going to explore how to move around the basic parabola, represented by the parent function f(x) = x², to get a new function, g(x). Our goal is to understand what kind of transformation is needed to go from f(x) = x² to g(x) = x² + 7. This stuff is super useful for understanding how graphs work and how changing the equation affects the picture. So, let's get started and make sense of this together!

Parent Function and Transformations: A Quick Review

Alright, first things first: the parent function. In this case, our parent function is f(x) = x². This is your basic, everyday parabola – a U-shaped curve that sits right in the middle of your graph, with its lowest point (the vertex) at the origin (0,0). Think of it as the original, the starting point for all our transformations. Now, what are transformations? Simply put, they are ways to alter the parent function. There are a few types: shifts (moving the graph), stretches and compressions (making the graph wider or narrower), and reflections (flipping the graph). Today, we're focusing on shifts.

Shifts are pretty straightforward: you move the entire graph either up, down, left, or right. The key is knowing how the equation changes to reflect these shifts. This is where things get interesting. If you add or subtract a number outside the function (like we're doing here: x² + 7), it's a vertical shift. Adding a number shifts the graph up, and subtracting shifts it down. If you add or subtract a number inside the function (e.g., (x + 7)²), that's a horizontal shift. Adding shifts it to the left, and subtracting shifts it to the right. It's a bit counterintuitive, but once you get the hang of it, it's a piece of cake! So, with this quick refresher, let's break down our specific problem.

To really get this, think about plotting some points. If you put x = 0 into f(x) = x², you get f(0) = 0. The point (0,0) is on the graph. Now, for g(x) = x² + 7, if you put x = 0, you get g(0) = 7. This means the point (0,7) is on the graph of g(x). See how it moved up? We'll get into more detail below, but this is the basic idea. We will also look at some examples so you can master this concept.

Deciphering g(x) = x² + 7

Now, let's get down to the nitty-gritty of g(x) = x² + 7. Remember, our parent function is f(x) = x². The key difference here is that we've added + 7 to the equation. This + 7 is outside the x² part. That means we're dealing with a vertical shift. Specifically, this means we are shifting the graph upwards by 7 units. Think about it: every single y-value on the graph of f(x) is being increased by 7. The vertex (0,0) of f(x) becomes the vertex (0,7) of g(x). Every other point on the graph moves up by the same amount.

Let's look at a few examples to solidify this understanding. If you take the point (1,1) on f(x), the corresponding point on g(x) will be (1,8), because g(1) = 1² + 7 = 8. Similarly, the point (-2,4) on f(x) becomes (-2,11) on g(x), because g(-2) = (-2)² + 7 = 11. So, the entire parabola has simply moved up. It’s a vertical translation. This kind of transformation doesn't change the shape of the parabola. It just moves it around on the coordinate plane. No stretching, shrinking, or flipping – just a straightforward upward shift. Keep in mind that the + 7 is added outside the square, which is critical to understanding that it's a vertical shift. If the + 7 were inside the square like this (x + 7)², it would be a horizontal shift.

To drive the point home, let’s visualize this. Imagine the original parabola f(x). Now, picture lifting the entire curve straight up by 7 units. That's g(x)! The shape is identical, but the position is different. The axis of symmetry (the vertical line that cuts the parabola in half) stays the same (x = 0), but the lowest point, the vertex, is now at (0,7). This upward shift is a fundamental concept in understanding how transformations work.

Eliminating the Incorrect Options

Now that we understand the correct transformation, let's look at the other options and why they're wrong. This is a great way to reinforce your understanding and avoid common mistakes.

• a. Shift left by 7 units: This is incorrect. A shift left would involve changing the equation to something like (x + 7)². The + 7 would be inside the parentheses, affecting the x-values. This is a horizontal shift, not a vertical one.
• b. Shift right by 7 units: Again, this is incorrect. A shift right would also involve a horizontal transformation. Similar to shifting left, the equation would look something like (x - 7)². The – 7 would be inside the parentheses, affecting the x-values.
• d. Shift down by 7 units: This is also wrong. A shift down would require subtracting 7 from the equation, making it g(x) = x² - 7. The – 7 would be outside the x², just like our original equation, but it would be a subtraction instead of an addition.

So, by process of elimination and by understanding the underlying concepts, we can easily see that the only correct option is a shift up by 7 units. Always pay close attention to where the change is happening in the equation—inside or outside the function. This distinction is crucial to understanding the type of transformation. Recognizing this pattern can simplify your approach to problems and help you to be successful in all areas of math.

Conclusion: Mastering Transformations

So, there you have it! Going from f(x) = x² to g(x) = x² + 7 involves a simple vertical shift up by 7 units. The graph of g(x) is the same parabola as f(x), but it's been moved upwards on the coordinate plane. This is a fundamental concept in understanding function transformations, and now you have a solid understanding. Remember, understanding the difference between adding or subtracting inside and outside the function is key to success. Keep practicing, and you'll become a transformation master in no time!

In short, when you add a constant outside the function, you're shifting vertically. Adding shifts up, and subtracting shifts down. When you add or subtract inside the function (affecting the x), you're shifting horizontally. Adding shifts left, and subtracting shifts right. Always pay attention to where the change is happening, and you'll be able to figure out the transformations like a pro. If you're still unsure, sketch out a quick graph. That will help you visualize the shifts. Keep up the awesome work!