Problem 80 - Precalculus

Question

In Exercises \(67-94,\) add the ordinates of the individual functions to graph each summed function on the indicated interval. $$y=\frac{1}{2} \sin x+2 \cos (4 x),-\pi \leq x \leq \pi$$

Step-by-Step Solution

Answer

Graph the function \( y = \frac{1}{2} \sin x + 2 \cos(4x) \) by adding ordinates of components over the interval \(-\pi \leq x \leq \pi\).

Step 1: Understanding the Function Components

The given function is the sum of two parts: \( \frac{1}{2} \sin x \) and \( 2 \cos(4x) \). We need to evaluate these functions separately and then add them together over the interval \(-\pi \leq x \leq \pi\).

Step 2: Evaluate and Graph \( \frac{1}{2} \sin x \)

Plot the function \( \frac{1}{2} \sin x \) over the given interval. The amplitude of this sine function is \( \frac{1}{2} \), and it completes one full cycle over the interval \(-\pi \leq x \leq \pi\).

Step 3: Evaluate and Graph \( 2 \cos(4x) \)

Plot the function \( 2 \cos(4x) \) over the interval. The amplitude of this cosine function is 2, but it completes four cycles since the argument of cosine is \( 4x \).

Step 4: Add the Ordinates

For each specific \( x \) value within \(-\pi \leq x \leq \pi\), calculate the corresponding \( y \)-value by summing the \( y \)-values from the \( \frac{1}{2} \sin x \) and \( 2 \cos(4x) \) functions.

Step 5: Graph the Summed Function

Using the computations from the previous step, plot the summed function \( y = \frac{1}{2} \sin x + 2 \cos(4x) \) over the interval \(-\pi \leq x \leq \pi\), by plotting the combined ordinates for each \( x \).

Key Concepts

sine function

The sine function, denoted as \( \sin x \), is one of the fundamental trigonometric functions. It captures the oscillatory nature of waves, making it crucial in various fields like physics and engineering. The basic form of the sine function is \( y = \sin x \), where \( y \) oscillates between -1 and 1.
In the exercise context, we deal with a modified sine function \( \frac{1}{2} \sin x \). This variation affects the amplitude but retains the same oscillatory nature. The sine wave completes one full cycle over the interval from \(-\pi\) to \(\pi\).
Key characteristics of the sine function:

• Starts at the origin \( (0, 0) \) in its basic form.
• Rises to its maximum value of \( +1 \) before returning to \( 0 \) and then dipping to \(-1\).
• The cycle repeats indefinitely, characteristic of periodic functions.

Understanding these attributes helps in visualizing and graphing transformations like scaling or shifts.

cosine function

The cosine function is another primary trigonometric function, symbolized as \( \cos x \). Like the sine function, it describes periodic oscillations, crucial for modeling wave behavior.
In the exercise, we encounter \( 2 \cos(4x) \). This indicates a modification where the amplitude is scaled and the frequency is increased, resulting in more cycles within a given interval.
The fundamental properties of the cosine function include:

• The function starts at its maximum value, \( y = 1 \), at \(x = 0\).
• It decreases to \(-1\) and back to \(1\) over one complete cycle.
• The modified frequency \(4x\) quadruples the number of cycles over the interval \(-\pi\) to \(\pi\).

Recognizing these traits allows for accurate plotting and comprehension of shifts in amplitude and frequency.

amplitude

Amplitude is a vital concept in understanding how high or low a trigonometric function reaches. For both sine and cosine functions, amplitude refers to the vertical distance from the midline to the peak of the wave.
The amplitude helps in determining the "loudness" or strength of the wave.
In the exercise:

• For \( \frac{1}{2} \sin x \), the amplitude is \( \frac{1}{2} \). This means the wave oscillates only up to 0.5 units above and below the midline.
• For \( 2 \cos(4x) \), the amplitude is 2, showing a more pronounced oscillation.

Amplitude does not affect the period or frequency but directly impacts the height of the wave. Understanding amplitude allows for a clearer perception of the wave's intensity.

periodic functions

A function is periodic if it repeats its values in regular intervals or periods. Trigonometric functions like sine and cosine are classic examples of periodic functions because they repeat their oscillations indefinitely.
This repetition is essential in modeling natural phenomena such as sound waves and tides.
The period for a standard sine or cosine function is \(2\pi\). However, in transformations, this can change:

• For \( \sin x \), the period remains \(2\pi\).
• For \( \cos(4x) \), the period reduces to \( \frac{\pi}{2} \) due to the frequency multiplication by 4.

Recognizing the periodicity helps in predicting the behavior of functions across different domains and intervals. It aids in precise plotting and application in real-world scenarios.