Maximum And Minimum Values Of F(x) = 4sin(3x) A Comprehensive Guide

Hey guys! Let's dive into the fascinating world of trigonometric functions and explore how to find the maximum and minimum values of a function. In this article, we'll specifically tackle the function f(x) = 4sin(3x). Don't worry, it's not as intimidating as it looks! We'll break it down step by step and make sure you understand not just the what, but also the why behind the solution.

Understanding the Sine Function

Before we jump into our specific function, let's refresh our understanding of the sine function itself. The sine function, denoted as sin(x), is a fundamental trigonometric function that oscillates between -1 and 1. This means that for any value of x, the output of sin(x) will always be within this range. This is a crucial piece of information for solving our problem. You can visualize this oscillation as a wave going up and down. The peaks of the wave reach a maximum value of 1, and the troughs reach a minimum value of -1. It's this periodic, wave-like behavior that gives the sine function its unique properties and makes it so useful in modeling various phenomena in physics, engineering, and even music!

The Sine Wave and its Bounds

Think of the sine function as a roller coaster. It goes up, it goes down, and it keeps repeating the same pattern. The highest point the roller coaster reaches is 1, and the lowest point is -1. It never goes higher or lower than these values. This is what we mean when we say the sine function oscillates between -1 and 1. Understanding this fundamental property is key to unlocking the secrets of trigonometric functions. When we analyze more complex functions involving sine, we can use this basic knowledge as a building block. For instance, knowing that sine is bounded between -1 and 1 allows us to predict the range of other functions that are derived from it, like our f(x) = 4sin(3x).

Visualizing the Sine Function

If you were to draw the graph of sin(x), you'd see a smooth, continuous wave that repeats itself infinitely. The x-axis represents the input values, and the y-axis represents the output values. The wave crosses the x-axis at multiples of π (pi), and it reaches its maximum value of 1 at x = π/2 + 2πk and its minimum value of -1 at x = 3π/2 + 2πk, where k is any integer. This visual representation can be incredibly helpful in understanding the behavior of the sine function. It allows us to see the cyclical nature of the function and how it oscillates between its maximum and minimum values. So, if you're ever feeling lost, try sketching the sine wave – it might just give you the insight you need!

Why This Matters for Our Problem

So, why are we talking so much about the sine function's bounds? Because it's the heart of our problem! The function we're analyzing, f(x) = 4sin(3x), is built upon the sine function. The 4 in front of the sine multiplies the entire function, and the 3x inside the sine affects the frequency of the oscillations. But the core behavior, the oscillation between -1 and 1, still comes from the sine function. By understanding how the sine function behaves, we can predict how these transformations will affect the maximum and minimum values of our function. This is a powerful concept in mathematics – breaking down complex problems into smaller, more manageable parts.

Analyzing f(x) = 4sin(3x)

Now, let's focus on our specific function, f(x) = 4sin(3x). This function takes the basic sine wave and modifies it in two key ways: it multiplies the amplitude by 4 and it compresses the period by a factor of 3. Let's explore each of these modifications and how they affect the maximum and minimum values.

The Amplitude: The Role of the '4'

The '4' in front of the sine function acts as an amplitude multiplier. What does that mean? Well, remember that the sine function normally oscillates between -1 and 1. Multiplying it by 4 stretches this range. Instead of oscillating between -1 and 1, our function now oscillates between -4 and 4. This means the maximum value of our function is 4, and the minimum value is -4. Think of it like stretching a rubber band. The '4' pulls the sine wave further up and further down, increasing its height (or amplitude). This is a direct and easily understandable effect.

To further illustrate, imagine plotting the graph of 4sin(x) versus sin(x). You'd see that the wave of 4sin(x) has the same shape as sin(x), but it's been vertically stretched by a factor of 4. The peaks are four times higher, and the troughs are four times lower. This vertical stretch is the essence of amplitude modification. Understanding amplitude is crucial in many applications, such as sound waves (where amplitude corresponds to loudness) and light waves (where amplitude corresponds to brightness).

The Period: The Role of the '3'

The '3' inside the sine function, within the argument 3x, affects the period of the function. The period of a trigonometric function is the length of one complete cycle. The standard sine function, sin(x), has a period of 2π. However, when we have sin(3x), the period is compressed. The new period is 2π/3. This means the function completes three full oscillations within the same interval that the standard sine function completes one. Think of it as squeezing the sine wave horizontally. The wave is squished together, making it oscillate faster.

To understand why the period changes, consider that the sine function completes one full cycle when its argument goes from 0 to 2π. So, for sin(3x), one full cycle occurs when 3x goes from 0 to 2π. Solving for x, we get x going from 0 to 2π/3. This confirms that the period is indeed 2π/3. This compression of the period affects how frequently the function reaches its maximum and minimum values. It will reach these values more often than the standard sine function.

Combining Amplitude and Period

So, we've seen how the '4' affects the amplitude and the '3' affects the period. Together, these transformations give us a function that oscillates between -4 and 4, and completes its cycles three times faster than the standard sine function. This understanding of amplitude and period is crucial for analyzing and interpreting the behavior of trigonometric functions. It allows us to predict the maximum and minimum values and how frequently they occur.

Finding the Maximum and Minimum Values

Now that we understand the transformations, let's pinpoint exactly where the maximum and minimum values occur for f(x) = 4sin(3x). We know the maximum value is 4 and the minimum value is -4. But at what values of x do these occur?

When Does the Maximum Occur?

The maximum value of f(x) = 4sin(3x) is 4. This occurs when sin(3x) reaches its maximum value, which is 1. So, we need to find the values of x for which sin(3x) = 1. We know that sin(θ) = 1 when θ = π/2 + 2πk, where k is any integer. In our case, θ = 3x. So, we have:

3x = π/2 + 2πk

Dividing both sides by 3, we get:

x = (π/6) + (2πk/3)

This formula gives us all the values of x where the function reaches its maximum value of 4. By plugging in different integer values for k (e.g., 0, 1, -1, 2, -2, etc.), we can find specific x-values where the maximum occurs. For example, when k = 0, x = π/6. When k = 1, x = (π/6) + (2π/3) = 5π/6. And so on.

When Does the Minimum Occur?

The minimum value of f(x) = 4sin(3x) is -4. This occurs when sin(3x) reaches its minimum value, which is -1. So, we need to find the values of x for which sin(3x) = -1. We know that sin(θ) = -1 when θ = 3π/2 + 2πk, where k is any integer. Again, θ = 3x. So, we have:

3x = 3π/2 + 2πk

Dividing both sides by 3, we get:

x = (π/2) + (2πk/3)

This formula gives us all the values of x where the function reaches its minimum value of -4. Just like with the maximum, we can plug in different integer values for k to find specific x-values. For example, when k = 0, x = π/2. When k = 1, x = (π/2) + (2π/3) = 7π/6. And so on.

Generalizing the Solution

We've found the formulas that give us all the x-values where the function reaches its maximum and minimum values. These formulas are powerful tools because they allow us to find these values for any cycle of the function. This ability to generalize solutions is a key aspect of mathematics. It allows us to make predictions and solve problems not just for a specific case, but for an entire class of cases.

Putting It All Together

So, to recap, we've successfully determined the maximum and minimum values of the function f(x) = 4sin(3x) and identified the intervals of x where these values occur. We found that:

• The maximum value is 4, and it occurs at x = (π/6) + (2πk/3), where k is any integer.
• The minimum value is -4, and it occurs at x = (π/2) + (2πk/3), where k is any integer.

We achieved this by understanding the fundamental properties of the sine function, how transformations like amplitude and period affect its behavior, and how to solve trigonometric equations. This process highlights the importance of building a strong foundation in basic concepts to tackle more complex problems.

I hope this breakdown has been helpful and has demystified the process of finding maximum and minimum values of trigonometric functions. Keep practicing, and you'll become a pro in no time! Remember, math is like a puzzle – it might seem challenging at first, but with the right tools and understanding, you can solve it!