Problem 90 - Precalculus

Question

In Exercises \(67-94,\) add the ordinates of the individual functions to graph each summed function on the indicated interval. $$y=2 \cos \left(\frac{3}{2} x\right)-\cos \left(\frac{1}{2} x\right),-2 \pi \leq x \leq 2 \pi$$

Step-by-Step Solution

Answer

Graph \( y = 2 \cos\left(\frac{3}{2}x\right) - \cos\left(\frac{1}{2}x\right) \) by adding the ordinates over \(-2\pi \leq x \leq 2\pi\).

Step 1: Identify the Individual Functions

We have the following functions: \( y_1 = 2 \cos\left(\frac{3}{2}x\right) \) and \( y_2 = -\cos\left(\frac{1}{2}x\right) \). We need to consider these two functions and understand that our goal is to add them together for the given range of \( x \).

Step 2: Determine the Summed Function

The summed function \( y \) is obtained by adding the individual ordinates: \( y = 2 \cos\left(\frac{3}{2}x\right) - \cos\left(\frac{1}{2}x\right) \). This equation represents the total vertical displacement at each point \( x \).

Step 3: Establish the Range

The problem asks us to consider the interval \( -2\pi \leq x \leq 2\pi \). This interval is from \(-2\pi\) to \(2\pi\) on the x-axis, which includes one complete cycle of each function given, with overlapping domains to account for their cycles.

Step 4: Evaluate and Graph the Functions

For each value of \( x \) within the interval \( -2\pi \leq x \leq 2\pi \), calculate \( y_1 \) and \( y_2 \) using the given trigonometric expressions. Add these values to evaluate the summed function \( y \). Plot the values of \( x \) versus \( y \) on a graph.

Step 5: Interpret the Summed Graph

The graph of the function \( y = 2\cos\left(\frac{3}{2}x\right) - \cos\left(\frac{1}{2}x\right) \) shows how the two cosine functions interfere with each other over the interval. Look for patterns, such as periodic peaks and troughs, to understand how the individual wave behaviors combine to form the overall shape of the graph.

Key Concepts

Cosine Function

The cosine function, one of the foundational trigonometric functions, plays a significant role in this problem. It helps to describe periodic phenomena such as waves and oscillations. The general form of the cosine function is \( y = A \cos(Bx + C) + D \). In this form,

• \( A \) represents the amplitude, which affects the height of the wave's peaks and troughs.
• \( B \) influences the period of the function, determining how frequently the wave repeats over a given interval.
• \( C \) represents the phase shift, which moves the wave horizontally.
• \( D \) is the vertical shift, raising or lowering the entire function.

In the exercise, we deal with two cosine functions: \( y_1 = 2 \cos\left(\frac{3}{2}x\right) \) and \( y_2 = -\cos\left(\frac{1}{2}x\right) \). Each of these functions showcases different amplitudes and frequencies due to their different coefficients.

Function Addition

Function addition involves combining two or more functions into a single function. In simple terms, this means for each value of \( x \), you add the corresponding \( y \)-values from each function. This concept allows us to understand complex wave interactions, often seen in engineering and physics.
In our exercise, we add two functions:

• \( y_1 = 2 \cos\left(\frac{3}{2}x\right) \), which is positive.
• \( y_2 = -\cos\left(\frac{1}{2}x\right) \), which is negative, introducing a subtraction into our calculation.

The resultant function is expressed as \( y = 2 \cos\left(\frac{3}{2}x\right) - \cos\left(\frac{1}{2}x\right) \). This summed function represents the net effect when these two trigonometric waves overlap. By calculating this function for each \( x \) value in the interval, we can observe how combining these waves influences the overall graph.

Periodic Functions

Periodic functions, like cosine, repeat their values at regular intervals. These functions are characterized by their periods, the distance along the x-axis before the function starts its pattern anew.
Considering the function \( y = 2 \cos\left(\frac{3}{2}x\right) - \cos\left(\frac{1}{2}x\right) \), it is a summation of two periodic functions:

• The first function \( 2 \cos\left(\frac{3}{2}x\right) \) has a smaller period compared to the second one because of the multiplier \( \frac{3}{2} \).
• The second function \(-\cos\left(\frac{1}{2}x\right) \) is less compressed, thus has a longer period.

By graphing these together over the interval \(-2\pi \leq x \leq 2\pi\), we visualize their interference patterns. This is typical of periodic functions: their interactions create complex waves, adding depth and dimensionality to the graph. Such periodic behavior is prevalent in natural phenomena and useful for modeling cycles in various fields.