Transforming Quadratics: From F(x) To G(x) – A Step-by-Step Guide

Hey guys, let's dive into some cool math! Today, we're figuring out how the graph of a simple quadratic function, f(x) = x², transforms into a more complex one, g(x) = 9x² - 36x. This isn't just about memorizing formulas; it's about understanding how graphs move and change. We will break it down into easy-to-digest steps. Believe me, it's way more fun than it sounds! We will be answering the question: Which transformation changes f(x) = x² to g(x) = 9x² - 36x?

Unveiling the Transformations

First, let's talk about the possible transformations. We have some options: whether the graph of f(x) = x² is widened, whether it shifts to the right, or whether it shifts up. Each of these changes the original parabola in a specific way. Our goal is to figure out which combination of these actions results in the graph of g(x) = 9x² - 36x. This will be like a little mathematical adventure, so buckle up!

To figure this out, we need to manipulate the equation g(x) = 9x² - 36x and rewrite it. The goal is to make it look similar to the original function to better understand the transformations. This will involve completing the square, a powerful technique that helps us rewrite quadratics into a form that reveals their transformations. The standard form of a quadratic equation is ax² + bx + c. The vertex form is a(x - h)² + k, where (h, k) is the vertex of the parabola. We want to convert our given equation into the vertex form to understand the shifts and stretches involved.

Now, let's get into the main task of completing the square. Taking g(x) = 9x² - 36x, we begin by factoring out the coefficient of the x² term (which is 9 in this case) from the first two terms. This gives us g(x) = 9(x² - 4x). Next, we need to figure out what value to add and subtract inside the parenthesis to complete the square. To do this, we take half of the coefficient of the x term (-4), square it, and then add it inside the parenthesis. Half of -4 is -2, and (-2)² is 4. Thus, we add and subtract 4 inside the parenthesis. Our equation now looks like this: g(x) = 9(x² - 4x + 4 - 4). Because we added 4 inside the parentheses, but the whole parentheses is multiplied by 9, we are essentially adding 36 (9 * 4). To keep the equation balanced, we also subtract 36 to get the equation to be g(x) = 9(x² - 4x + 4) - 36. By doing this, we completed the square to then factor it. It is also important to note that we can add and subtract the same value without changing the equation's value. Then, we can factor the perfect square trinomial inside the parenthesis, which becomes (x - 2)². Therefore, our equation becomes g(x) = 9(x - 2)² - 36.

Analyzing the Vertex Form

Now that we have g(x) = 9(x - 2)² - 36 in vertex form, we can clearly see the transformations. The number '9' in front of the (x - 2)² part indicates a vertical stretch by a factor of 9. This means the graph will be narrower, not wider, thus eliminating option A. The -2 inside the parenthesis indicates a horizontal shift to the right by 2 units. Finally, the -36 at the end indicates a vertical shift down by 36 units. As we can see, the equation g(x) = 9(x - 2)² - 36 demonstrates a horizontal shift to the right by 2 units, a vertical stretch, and a vertical shift downwards.

Deep Dive into Transformations

Alright, let's take a closer look at each type of transformation. We have already explored the horizontal shift and the vertical stretch, but it's important to understand each one in detail. When we are talking about a horizontal shift, it means the entire graph is moved left or right. A shift to the right by 'h' units means subtracting 'h' from the x inside the function, as we saw with the (x - 2) term. This directly influences the location of the parabola's vertex. The vertical stretch, which is represented by a factor multiplied by the function, changes the steepness of the graph. If the factor is greater than 1, like the 9 in our example, the graph stretches vertically, becoming narrower. If the factor is between 0 and 1, the graph compresses vertically, becoming wider. In our case, the stretch by a factor of 9 dramatically changes the shape.

Now, let's look at the vertical shift. This moves the entire graph up or down along the y-axis. The '- 36' at the end of our equation shifts the graph down by 36 units. This is straightforward: a positive value shifts the graph up, and a negative value shifts it down. So, when the graph is in vertex form, a(x - h)² + k, the 'h' affects the horizontal shift, and the 'k' affects the vertical shift. These combined transformations change the vertex of the parabola from the origin, in the case of f(x) = x², to the point (2, -36) for g(x) = 9x² - 36x.

The Correct Answer and Explanation

Based on our analysis, we know that option B is the correct answer: The graph of f(x) = x² is shifted right 2 units. This is because the equation g(x) = 9(x - 2)² - 36 tells us exactly this transformation. The number inside the parenthesis, (x - 2), indicates a horizontal shift. The negative sign signifies a shift to the right. The fact that the graph is shifted to the right, also indicates that the graph is narrower, which is due to the vertical stretch factor. The correct answer is B, the graph of f(x) = x² is shifted right 2 units. The graph also undergoes a vertical stretch and a vertical shift.

Therefore, understanding the vertex form allows us to accurately determine the transformations involved. So, when you look at a quadratic equation, remember the key elements: the vertical stretch, the horizontal shift, and the vertical shift. They work together to change the appearance and position of the parabola. Mastering these transformations is key to understanding and solving complex problems.

Conclusion: Mastering Quadratic Transformations

So there you have it, guys! We've successfully transformed f(x) = x² into g(x) = 9x² - 36x by completing the square and identifying the key transformations. We saw a horizontal shift, a vertical stretch, and a vertical shift. Keep practicing, and you'll become a pro at this. Remember, math is all about practice and understanding. The more you work with these equations and graphs, the easier it will become. Keep up the great work, and you'll be acing those math problems in no time! Keep exploring and have fun with it!