Finding Corresponding Points Function Transformation F(-3/4 X)

Hey everyone! Today, we're diving into the world of function transformations. Specifically, we're going to tackle a problem where we need to find the corresponding point on a transformed function given a point on the original function. It might sound a bit complicated, but trust me, we'll break it down step by step. Let's get started!

The Basics of Function Transformations

Before we jump into the problem, let's quickly recap the basics of function transformations. Function transformations are ways to modify the graph of a function by shifting, stretching, compressing, or reflecting it. These transformations are applied to the input (x-values) or the output (y-values) of the function, resulting in a new graph that is related to the original graph.

Horizontal Transformations: These transformations affect the x-values of the function. They include horizontal shifts (left or right) and horizontal stretches or compressions. When we modify the input of a function, like replacing x with ax, we're dealing with a horizontal transformation. For example, f(ax) compresses the graph horizontally if |a| > 1 and stretches it if 0 < |a| < 1. If a is negative, it also reflects the graph across the y-axis.

Vertical Transformations: These transformations affect the y-values of the function. They include vertical shifts (up or down) and vertical stretches or compressions. When we modify the output of a function, like adding a constant or multiplying by a constant, we're dealing with a vertical transformation. For instance, af(x) stretches the graph vertically if |a| > 1 and compresses it if 0 < |a| < 1. If a is negative, it also reflects the graph across the x-axis.

In our problem, we're focusing on a horizontal transformation, specifically how the input x is being modified. This will help us understand how the points on the graph of the function change.

Problem Setup: Finding the Corresponding Point

Okay, let's get to the heart of the problem. We're given that the point (-3, 7) lies on the graph of the function f(x). This means that when x = -3, f(x) = 7. In other words, f(-3) = 7. We're then asked to find the corresponding point on the graph of the transformed function f(-3/4 x).

The key here is to figure out how the x-value changes under the transformation. The transformation f(-3/4 x) involves both a horizontal stretch/compression and a reflection. The -3/4 factor inside the function is what's causing this transformation. To find the corresponding point, we need to determine what x-value in the transformed function will give us the same input to the original function f as we had before (which was -3).

So, we need to solve the equation -3/4 x = -3. This will tell us what x-value we need to plug into f(-3/4 x) to get the same result as f(-3). Once we find this x-value, the y-value will remain the same since we haven't applied any vertical transformations.

Solving for the New x-value

Let's solve the equation -3/4 x = -3. To isolate x, we can multiply both sides of the equation by the reciprocal of -3/4, which is -4/3. This gives us:

x = -3 * (-4/3)

x = 4

So, the new x-value is 4. This means that f(-3/4 * 4) = f(-3) = 7. The y-value remains the same because the transformation only affects the x-values. There is no vertical stretching, compression, or reflection happening in this case.

Therefore, the corresponding point on the graph of f(-3/4 x) is (4, 7). This is the key to understanding how horizontal transformations work. By setting the transformed input equal to the original input, we can find the new x-coordinate that corresponds to the same y-coordinate.

Putting It All Together: Step-by-Step Solution

Let's recap the steps we took to solve this problem. This will help solidify the process in your mind and make it easier to tackle similar problems in the future.

1. Identify the Given Information: We were given the point (-3, 7) on the graph of f(x), which means f(-3) = 7. We were also given the transformed function f(-3/4 x).
2. Understand the Transformation: The transformation f(-3/4 x) involves a horizontal stretch/compression and a reflection due to the -3/4 factor. We need to find the new x-value that corresponds to the same y-value.
3. Set Up the Equation: We set the transformed input equal to the original input: -3/4 x = -3. This equation represents the condition where the input to the function f remains the same after the transformation.
4. Solve for x: We solved the equation for x by multiplying both sides by -4/3, which gave us x = 4.
5. Determine the Corresponding Point: Since the y-value remains the same, the corresponding point on the graph of f(-3/4 x) is (4, 7).

By following these steps, you can confidently find corresponding points for various function transformations. The key is to focus on how the input (x-value) is being modified and solve for the new x-value that gives you the same input to the original function.

Visualizing the Transformation

To further enhance your understanding, it's helpful to visualize what's happening with the graph. The transformation f(-3/4 x) does two things to the graph of f(x):

1. Horizontal Stretch/Compression: The factor of 3/4 inside the function causes a horizontal stretch. Since 3/4 is between 0 and 1, the graph is stretched horizontally by a factor of 4/3 (the reciprocal of 3/4).
2. Reflection across the y-axis: The negative sign in -3/4 x reflects the graph across the y-axis. This means that points on the left side of the y-axis in the original graph will move to the right side in the transformed graph, and vice versa.

So, the point (-3, 7) is first stretched horizontally by a factor of 4/3, which would move it to (-4, 7). Then, it's reflected across the y-axis, which moves it to (4, 7). This visual representation confirms our algebraic solution.

Common Mistakes to Avoid

When dealing with function transformations, it's easy to make a few common mistakes. Let's go over some of these so you can avoid them:

• Forgetting the Order of Transformations: When multiple transformations are applied, the order matters. In general, horizontal stretches/compressions and reflections should be applied before horizontal shifts. Similarly, vertical stretches/compressions and reflections should be applied before vertical shifts.
• Incorrectly Applying Horizontal Transformations: Horizontal transformations are often counterintuitive. For example, f(ax) compresses the graph horizontally if |a| > 1 and stretches it if 0 < |a| < 1. It's easy to mix this up, so always double-check the effect of the transformation.
• Ignoring the Negative Sign: A negative sign inside the function (like in our problem) indicates a reflection across the y-axis. Don't forget to account for this reflection when finding corresponding points.
• Assuming the y-value Changes Automatically: The y-value only changes if there's a vertical transformation. In our problem, since we only had a horizontal transformation, the y-value remained the same.

By being aware of these common mistakes, you can approach function transformation problems with more confidence and accuracy.

Practice Problems

To master function transformations, practice is key. Here are a few practice problems you can try:

1. Given that (2, -5) is on the graph of f(x), find the corresponding point for the function f(2x).
2. Given that (-1, 3) is on the graph of f(x), find the corresponding point for the function f(-x).
3. Given that (4, 1) is on the graph of f(x), find the corresponding point for the function f(1/2 x).

Work through these problems, and you'll become more comfortable with identifying and applying function transformations. Remember to focus on how the x-value changes and set up the appropriate equation to solve for the new x-coordinate.

Conclusion: Mastering Function Transformations

Function transformations are a fundamental concept in mathematics. Understanding how they work allows you to analyze and manipulate graphs of functions with ease. By breaking down the transformations step by step and visualizing their effects, you can confidently find corresponding points and solve a wide range of problems.

In this article, we tackled a specific problem involving a horizontal stretch/compression and a reflection. We learned how to set up an equation to find the new x-value and how to determine the corresponding point on the transformed graph. Remember to pay attention to the order of transformations and avoid common mistakes.

Keep practicing, and you'll become a pro at function transformations! Now, go out there and transform your understanding of functions!