Amplitude, Period, And Maximum Value Of Trigonometric Functions

Hey guys! Let's dive into some awesome trigonometric function problems. We're going to tackle how to find the amplitude, period, and maximum values, and even sketch a graph. So, buckle up, and let's get started!

1. Finding Amplitude and Period: f(x) = 4 sin 3x - 2

Alright, so the first question we've got is about the function f(x) = 4 sin 3x - 2. We need to figure out its amplitude and period. These are super important characteristics that tell us a lot about the behavior of the sine function.

Understanding Amplitude

Let's start with the amplitude. Think of the amplitude as the height of the sine wave from its midline. In the general form of a sine function, f(x) = A sin(Bx + C) + D, the amplitude is simply the absolute value of A. Why absolute value? Because amplitude is a distance, and distances are always positive! This is a crucial concept to grasp.

In our case, f(x) = 4 sin 3x - 2, the value of A is 4. So, the amplitude is |4| = 4. Easy peasy, right? This means our sine wave will oscillate 4 units above and 4 units below its midline.

Calculating the Period

Next up is the period. The period tells us how long it takes for the sine wave to complete one full cycle. Imagine tracing the sine wave with your finger – the period is the distance you cover before the pattern repeats itself. For the general form f(x) = A sin(Bx + C) + D, the period is calculated using the formula: Period = 2π / |B|. Remember, π (pi) is that magical number approximately equal to 3.14159.

Now, in our function f(x) = 4 sin 3x - 2, the value of B is 3. Plugging this into our formula, we get: Period = 2π / |3| = 2π / 3. That's it! The period of our function is 2π / 3. This means the sine wave completes one full cycle in an interval of 2π / 3 units along the x-axis. Understanding the period is essential for graphing and analyzing trigonometric functions.

So, to recap, for f(x) = 4 sin 3x - 2, the amplitude is 4, and the period is 2π / 3. Knowing these values helps us visualize and understand the behavior of the function.

2. Finding the Maximum Value: f(x) = 3 - sin 3x

Now, let's switch gears and find the maximum value of the function f(x) = 3 - sin 3x. To tackle this, we need to remember the range of the sine function. The sine function, no matter what's inside the parentheses, always oscillates between -1 and 1. This is a fundamental property of sine waves.

Leveraging Sine's Range

So, sin 3x will always be between -1 and 1. That means the smallest value sin 3x can be is -1, and the largest value is 1. Now, let's plug these values into our function and see what happens.

When sin 3x = -1, we have: f(x) = 3 - (-1) = 3 + 1 = 4. This is looking promising for a maximum value!

On the other hand, when sin 3x = 1, we get: f(x) = 3 - 1 = 2. This is clearly smaller than 4.

Identifying the Maximum

Therefore, the maximum value of the function f(x) = 3 - sin 3x is 4. We found this by understanding the range of the sine function and plugging in the extreme values. This method is super useful for finding maximum and minimum values of trigonometric functions.

3. Sketching the Graph: f(x) = -cos 4x for 0 ≤ x ≤ π

Okay, guys, let's get graphical! We need to sketch the graph of the function f(x) = -cos 4x within the interval 0 ≤ x ≤ π. This might sound intimidating, but we'll break it down step-by-step.

Understanding the Cosine Function

First, let's recall the basic shape of the cosine function, cos x. It starts at its maximum value of 1, goes down to 0 at π / 2, reaches its minimum of -1 at π, goes back to 0 at 3π / 2, and finally returns to 1 at 2π. It's a smooth, wave-like curve. This basic understanding is key.

The Impact of Transformations

Now, what does the - sign in front of cos 4x do? It flips the graph vertically. So, instead of starting at 1, our graph will start at -1. The negative sign acts like a vertical reflection.

And what about the 4 inside the cosine function? It affects the period. Remember our period formula? Period = 2π / |B|. In this case, B = 4, so the period is 2π / 4 = π / 2. This means our cosine wave will complete one full cycle in an interval of π / 2, which is much faster than the standard cosine function.

Putting It All Together

We know our graph starts at -1, completes a full cycle in π / 2, and we're only interested in the interval 0 ≤ x ≤ π. This means we'll see two full cycles of the cosine wave within our interval.

To sketch the graph accurately, let's identify some key points:

• At x = 0, f(x) = -cos(4 * 0) = -cos(0) = -1.
• At x = π / 8, f(x) = -cos(4 * π / 8) = -cos(π / 2) = 0.
• At x = π / 4, f(x) = -cos(4 * π / 4) = -cos(π) = -(-1) = 1.
• At x = 3π / 8, f(x) = -cos(4 * 3π / 8) = -cos(3π / 2) = 0.
• At x = π / 2, f(x) = -cos(4 * π / 2) = -cos(2π) = -1.

And so on... We can continue this pattern to sketch the graph over the interval 0 ≤ x ≤ π. Remember, the graph will oscillate between -1 and 1, and it will complete two full cycles within the given interval.

Final Thoughts on Graphing

Sketching trigonometric functions might seem tricky at first, but with practice and a solid understanding of amplitude, period, and phase shifts, you'll become a pro in no time! This is critical for visualizing function behavior.

So, there you have it, guys! We've tackled amplitude, period, maximum values, and graph sketching for trigonometric functions. Keep practicing, and you'll ace those math problems!