Hey guys! Let's dive into the fascinating world of function transformations, specifically focusing on how the function f(x) = x² behaves when we tweak its input. This is a super important concept in mathematics, and grasping it will help you tackle a whole bunch of problems. We'll explore how different operations on the input x affect the graph of the quadratic function, making it wider, narrower, or shifting it horizontally. So, buckle up and let's unravel these transformations together!
The Foundation: f(x) = x²
Before we jump into the transformations, let's solidify our understanding of the base function, f(x) = x². This is a classic parabola, symmetrical about the y-axis, with its vertex (the lowest point) at the origin (0,0). The key here is to recognize how the squaring operation affects the x-values. Positive and negative x-values yield positive y-values, creating the characteristic U-shape. As x moves away from zero in either direction, y increases rapidly. This basic understanding is crucial for visualizing how transformations alter this fundamental shape. Think of this as your starting point – the template upon which we'll apply various changes. Without a strong grasp of f(x) = x², the transformations will seem more abstract and less intuitive. We need to know what we're changing from to fully understand the what and why of the changes themselves. So, take a moment to picture the parabola, its symmetry, and its rate of growth. This mental image will be your guide as we move forward.
Horizontal Transformations: Scaling the Input
Now, let's get to the heart of the matter: how do we transform f(x) = x²? We'll start by focusing on horizontal transformations, which involve directly manipulating the input, x. The options presented, g(x) = f(2x), g(x) = f(4x), and g(x) = f(1/2 x), all fall into this category. These transformations scale the x-values before they are squared. This might sound a bit abstract, so let's break it down. When we replace x with 2x, we're essentially compressing the graph horizontally. Think about it: to achieve the same y-value as f(x), g(x) = f(2x) only needs half the x-value. This results in a narrower parabola. Similarly, g(x) = f(4x) compresses the graph even further, making it even skinnier. On the flip side, g(x) = f(1/2 x) stretches the graph horizontally. To get the same y-value, g(x) now needs twice the x-value, resulting in a wider parabola. It's crucial to remember that these transformations act opposite to what you might initially expect. Multiplying x by a constant greater than 1 compresses the graph, while multiplying by a constant between 0 and 1 stretches it. Visualizing this inverse relationship is key to mastering horizontal transformations. Consider specific points: for f(x) = x², the point (2, 4) lies on the graph. For g(x) = f(2x) = (2x)², the point (1, 4) lies on the graph. See how the x-value has been halved, compressing the graph horizontally?
Analyzing Option A: g(x) = f(2x)
Let’s examine option A, g(x) = f(2x), more closely. As we discussed, this transformation horizontally compresses the graph of f(x) = x². Substituting 2x into the function, we get g(x) = (2x)² = 4x². This equation reveals that the y-values of g(x) are four times the y-values of f(x) for the same x-value. This confirms the compression, but it also introduces a vertical stretch. However, the primary effect is horizontal compression. Imagine taking the original parabola and squeezing it inwards from both sides – that's the effect of f(2x). The larger the coefficient of x inside the function, the greater the compression. So, f(3x) would be even narrower than f(2x). The key takeaway here is that multiplying x by a constant inside the function has an inverse effect on the horizontal dimension. This is a fundamental principle in function transformations, and understanding it will allow you to predict the behavior of various functions under similar manipulations. Think about how this compression affects the key features of the parabola, such as its vertex and axis of symmetry. The vertex remains at (0, 0), but the parabola becomes steeper and narrower.
Analyzing Option B: g(x) = f(4x)
Option B, g(x) = f(4x), takes the horizontal compression even further. Replacing x with 4x in f(x) = x² gives us g(x) = (4x)² = 16x². This means that for a given x-value, the corresponding y-value on g(x) is sixteen times larger than the y-value on f(x). This represents a significant horizontal compression and a corresponding vertical stretch. The graph of g(x) will be much narrower than the graph of f(x). To visualize this, imagine squeezing the parabola inwards even more than we did with f(2x). The effect is a sharper, more pointed curve. This further reinforces the inverse relationship between the coefficient of x inside the function and the horizontal stretch or compression. The larger the coefficient, the greater the compression. It's helpful to compare f(2x) and f(4x) directly to see the difference in their compression. For example, consider the point (1, 1) on f(x) = x². On f(2x), the corresponding y-value of 1 occurs at x = 1/2, and on f(4x), it occurs at x = 1/4. This clearly demonstrates the increased compression as the coefficient of x increases.
Analyzing Option D: g(x) = f(1/2 x)
Now, let's switch gears and examine option D, g(x) = f(1/2 x). This transformation introduces a horizontal stretch. When we replace x with 1/2 x, we get g(x) = (1/2 x)² = 1/4 x². This means that the y-values of g(x) are one-fourth the y-values of f(x) for the same x-value. This results in a wider parabola. Think of it as pulling the parabola outwards from both sides. To achieve the same y-value, g(x) needs a larger x-value compared to f(x). This stretching effect is the opposite of the compression we saw in options A and B. It's crucial to remember that multiplying x by a fraction between 0 and 1 stretches the graph horizontally. This can sometimes be counterintuitive, so it's helpful to visualize it. Imagine taking the original parabola and gently pulling it outwards – that's the effect of f(1/2 x). The smaller the fraction, the greater the stretch. So, f(1/4 x) would be even wider than f(1/2 x). Consider specific points: for f(x) = x², the point (2, 4) lies on the graph. For g(x) = f(1/2 x) = (1/2 x)², the point (4, 4) lies on the graph. See how the x-value has been doubled, stretching the graph horizontally?
Vertical Transformations: Scaling the Output
Option C, g(x) = 2f(x), represents a vertical transformation. This transformation affects the output of the function, f(x), rather than the input, x. Multiplying the entire function by a constant stretches or compresses the graph vertically. In this case, g(x) = 2f(x) = 2x². This means that for every x-value, the corresponding y-value on g(x) is twice the y-value on f(x). This results in a vertical stretch, making the parabola appear taller and narrower. Imagine taking the original parabola and pulling it upwards – that's the effect of 2f(x). Vertical transformations are generally more intuitive than horizontal transformations. Multiplying the function by a constant greater than 1 stretches the graph vertically, while multiplying by a constant between 0 and 1 compresses it. For example, 3f(x) would be stretched even more than 2f(x), while 1/2 f(x) would be compressed. It's important to distinguish between horizontal and vertical transformations. Horizontal transformations affect the x-values, while vertical transformations affect the y-values. This distinction is crucial for accurately predicting the behavior of transformed functions.
The Correct Equation: Understanding the Question
The question asks which equation represents function g. We need to consider what kind of transformation is implied by the question. Without additional context, it's difficult to definitively say which equation
