Transformed Tangent Function Equation: Period 10π

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Hey guys! Let's dive into a super interesting problem today that involves transforming trigonometric functions, specifically the tangent function. We're going to figure out how the equation changes when we stretch the graph of y = tan(x) horizontally, making its period 10π. This is a classic problem that blends graphical transformations with understanding the properties of trig functions. So, buckle up, and let's get started!

Understanding the Original Tangent Function: y = tan(x)

Before we jump into the transformation, let's quickly recap the basics of the tangent function, y = tan(x). This function has a period of π, which means its graph repeats itself every π units along the x-axis. Imagine the graph – it has vertical asymptotes at x = π/2 + nπ, where n is an integer, and it wiggles its way between these asymptotes. Think of it like a repeating wave, but instead of smooth crests and troughs, it has these sharp vertical lines that it never quite touches.

Key characteristics of y = tan(x) to remember:

  • Period: π
  • Vertical Asymptotes: x = π/2 + nπ, where n is an integer
  • Basic Shape: Repeats every π units, increasing from negative infinity to positive infinity between asymptotes.

Understanding these basics is crucial because when we stretch or compress the graph horizontally, we're essentially changing the period. And changing the period means changing the equation. Think of it like adjusting the zoom on a camera – you're changing how much of the scene fits into the frame, but the scene itself remains the same. Similarly, we're stretching the tangent function, but the fundamental wiggling behavior remains, just over a wider interval.

Horizontal Stretching and the Period

Now, let's talk about the horizontal stretch. When we horizontally stretch a function, we're essentially pulling it wider along the x-axis. In terms of the graph, this means the features of the graph – the peaks, valleys, and in this case, the asymptotes – are further apart. The period, which is the distance it takes for the graph to complete one full cycle, becomes longer.

In our problem, the period of y = tan(x) is stretched from π to 10π. That's a pretty significant stretch! So, how does this affect the equation? This is where the magic of function transformations comes in. Remember, horizontal transformations are a bit counterintuitive. To stretch the graph horizontally, we actually need to multiply the x-value inside the function by a fraction (or divide by a whole number). This is because we're essentially telling the function to take its values more slowly, thus stretching it out.

The general form for a horizontal stretch or compression of a function y = f(x) is y = f(Bx), where B is a constant. If |B| < 1, the graph is stretched horizontally. If |B| > 1, the graph is compressed horizontally. The new period is given by the original period divided by |B|. In our case, we need the new period to be 10π, and the original period is π. So, we have:

New Period = Original Period / |B|

10π = π / |B|

|B| = π / 10π

|B| = 1/10

Since we're stretching, B will be positive. So, B = 1/10. This means our transformed function will look something like y = tan(x/10).

Finding the Equation of the Transformed Function

Based on our understanding of horizontal stretching and the period, we've deduced that the transformed function should have the form y = tan(Bx), where B = 1/10. Let's plug this into our equation:

y = tan((1/10)x)

This simplifies to:

y = tan(x/10)

So, the equation of the transformed function is indeed y = tan(x/10). This makes intuitive sense – we've divided x by 10, which means the function will take 10 times longer to complete one cycle, thus stretching the period to 10π. Think about it like this: the original function completes a cycle in π units. To make it complete a cycle in 10π units, we need to slow down the input by a factor of 10, hence the division by 10 inside the tangent function.

To further solidify our understanding, let's consider what would happen to the vertical asymptotes. The original tangent function has asymptotes at x = π/2 + nπ. For the transformed function, the argument of the tangent function, x/10, must equal π/2 + nπ at the asymptotes. So, we have:

x/10 = π/2 + nπ

Multiply both sides by 10:

x = 5π + 10nπ

These new asymptotes are spaced 10π units apart, confirming that the period has indeed been stretched to 10π.

Analyzing the Answer Choices

Now that we've derived the equation of the transformed function, y = tan(x/10), let's take a look at the answer choices provided in the original problem. We should see that one of the options matches our result perfectly.

  • A. y = tan(x/10)
  • B. y = tan(x/5)
  • C. y = tan(5x)

Clearly, option A, y = tan(x/10), is the correct answer. The other options represent different transformations. Option B, y = tan(x/5), would represent a horizontal stretch, but only to a period of 5π. Option C, y = tan(5x), represents a horizontal compression, where the period would be π/5.

Common Mistakes and How to Avoid Them

Transformations can be tricky, and there are a few common mistakes people often make when dealing with horizontal stretches and compressions. Let's go over them so you can avoid these pitfalls!

  1. Confusing Horizontal and Vertical Transformations: It's easy to mix up horizontal and vertical stretches/compressions. Remember, horizontal transformations affect the x-values and are counterintuitive – a stretch involves dividing x by a number (or multiplying by a fraction), while a compression involves multiplying x by a number. Vertical transformations, on the other hand, affect the y-values and are more straightforward.
  2. Incorrectly Calculating the Stretch Factor: When calculating the factor by which the period changes, make sure you set up the equation correctly. Remember that the new period is equal to the original period divided by the absolute value of the transformation factor (|B|). So, if you're given the new period and need to find the factor, rearrange the equation accordingly.
  3. Forgetting the Absolute Value: The absolute value is crucial when dealing with horizontal stretches and compressions because it ensures that we're only considering the magnitude of the transformation, not the direction. A negative sign would indicate a reflection, which is a different type of transformation altogether.
  4. Not Checking the Asymptotes: For tangent and cotangent functions, checking the asymptotes is a great way to verify your answer. If you've stretched or compressed the graph horizontally, the asymptotes will shift accordingly. Make sure the asymptotes of your transformed function match the new period.

Practice Problems

To really master horizontal stretches and compressions, practice is key! Here are a couple of similar problems you can try:

  1. The graph of the function y = tan(x) was horizontally stretched so that its period became 6π. Which is the equation of the transformed function?
  2. The graph of the function y = tan(x) is transformed into y = tan(x/3). What is the period of the transformed function?

Work through these problems step-by-step, just like we did in the example. Pay close attention to how the period changes and how that affects the equation. If you get stuck, go back and review the concepts we covered earlier.

Conclusion

So, guys, we've successfully navigated the world of tangent function transformations! We started by understanding the basics of y = tan(x), then delved into horizontal stretching and how it affects the period. We figured out how to find the equation of the transformed function and even looked at some common mistakes to avoid. Remember, the key to mastering these transformations is to understand the relationship between the graph and the equation and to practice, practice, practice! Keep exploring, keep questioning, and keep those math skills sharp! You've got this!