Finding G(x) After Translating F(x): A Step-by-Step Guide

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Hey guys! Let's dive into a fun math problem today that involves translating trigonometric functions. We've got a function, f(x), and we're going to move it around on the graph to create a new function, g(x). This might sound tricky, but I'm here to break it down and make it super easy to understand. So, grab your pencils, and let's get started!

Understanding the Original Function f(x)

Okay, so first things first, let's take a good look at our original function: f(x) = 9cos(x - π/2) + 3. Breaking this down, we can identify the different transformations applied to the basic cosine function. The key here is understanding how each part of the equation affects the graph. This involves recognizing the amplitude, phase shift, and vertical shift.

  • Amplitude: The '9' in front of the cosine function represents the amplitude. What does that mean? Well, it tells us how tall the wave is – in this case, the graph stretches 9 units above and below its midline. The amplitude significantly impacts the function's range, which is crucial for visualizing and understanding the graph. So, remember that the amplitude stretches or compresses the graph vertically.
  • Phase Shift: Next up, we've got the '(x - Ï€/2)' inside the cosine. This indicates a phase shift, which is a horizontal translation. The '- Ï€/2' tells us the graph has been shifted Ï€/2 units to the right. Understanding phase shifts is essential for accurately graphing trigonometric functions. It's like moving the entire wave left or right along the x-axis, and it's a crucial component in many real-world applications, such as modeling periodic phenomena like sound waves or electrical signals. So, when you see something like (x - c), it means a shift to the right by c units.
  • Vertical Shift: Finally, we have the '+ 3' at the end. This is a vertical shift, which moves the entire graph up by 3 units. The vertical shift changes the midline of the function, and it's super important for identifying the function's range and equilibrium position. The vertical shift is one of the fundamental transformations to understand. It’s as simple as moving the whole graph up or down.

So, to recap, our original function f(x) is a cosine wave that's been stretched vertically by a factor of 9, shifted π/2 units to the right, and then shifted 3 units up. Got it? Great! Now, let's see what happens when we translate it even further.

Translating the Function to Create g(x)

Now comes the fun part – translating f(x) to get g(x). We're told that we need to shift f(x) by π/6 units to the left and 4 units up. Let’s tackle each shift one at a time to keep things clear and easy.

Shifting π/6 Units to the Left

When we shift a function to the left, we're dealing with a horizontal translation. But here’s the catch: to shift a function to the left, we actually add to the x-value inside the function. This might seem counterintuitive, but think of it this way: to get the same y-value, we need to input a smaller x-value, which means we need to add to the current x. So, to shift f(x) π/6 units to the left, we replace 'x' with '(x + π/6)' in the cosine part of the equation.

So far, our function looks like this: 9cos((x + π/6) - π/2) + 3. Notice how we've added π/6 inside the cosine function. This is the key to shifting the graph leftward. Now, we can simplify the expression inside the cosine to make it a bit cleaner. We have (x + π/6 - π/2). To combine these, we need a common denominator, which is 6. So, π/2 becomes 3π/6. Thus, we have (x + π/6 - 3π/6), which simplifies to (x - 2π/6), or even further to (x - π/3). So, after shifting π/6 units to the left, our function looks like 9cos(x - π/3) + 3. We're getting closer to g(x)!

Shifting 4 Units Up

Okay, now for the vertical shift. This one's a bit more straightforward. To shift the function 4 units up, we simply add 4 to the entire function. That’s it! No tricks here. So, we take our function from the previous step, 9cos(x - π/3) + 3, and add 4 to it.

This gives us 9cos(x - π/3) + 3 + 4, which simplifies to 9cos(x - π/3) + 7. And there you have it! We've successfully shifted the function π/6 units to the left and 4 units up. This new function is g(x).

The Final Equation for g(x)

So, after all that shifting and simplifying, we've arrived at the equation for g(x). Let's write it out clearly: g(x) = 9cos(x - π/3) + 7. This is the function we get after translating f(x) as described. Make sure you understand each step we took to get here. Understanding these transformations is super useful in many areas of math and science.

Common Mistakes to Avoid

Before we wrap up, let's quickly chat about some common mistakes people make when dealing with translations of functions. Knowing these pitfalls can save you some headaches down the road.

  • Incorrect Direction for Horizontal Shifts: One of the most common mistakes is getting the direction of the horizontal shift wrong. Remember, adding inside the function shifts the graph to the left, and subtracting shifts it to the right. It's the opposite of what you might intuitively think, so be careful! This is a crucial point to remember for any horizontal transformation.
  • Applying Shifts in the Wrong Order: The order in which you apply transformations matters. In general, horizontal shifts should be applied before vertical shifts. This is because horizontal changes affect the input (x-value), while vertical changes affect the output (y-value). Getting the order wrong can lead to an incorrect final equation. So, keep the order straight to ensure accuracy.
  • Forgetting to Simplify: After applying the shifts, don't forget to simplify the equation. This often involves combining constants or simplifying expressions inside the trigonometric functions. Simplifying makes the equation easier to work with and helps prevent errors in further calculations. Always take that extra step to simplify your result.

Wrapping Up

So there you have it! We've successfully found the equation for g(x) by translating f(x). Remember, the key is to understand how each transformation affects the graph and to apply them in the correct order. We tackled horizontal and vertical shifts, and we even talked about some common mistakes to watch out for. I hope this explanation has made things clearer for you.

Keep practicing these types of problems, and you'll become a translation master in no time! If you found this guide helpful, give it a thumbs up, and let me know in the comments if you have any other math topics you'd like me to cover. Happy calculating, guys!