Problem 89 - Precalculus

Question

In Exercises \(67-94,\) add the ordinates of the individual functions to graph each summed function on the indicated interval. $$y=-\frac{1}{4} \cos \left(\frac{\pi}{6} x\right)-\frac{1}{2} \cos \left(\frac{\pi}{3} x\right), 0 \leq x \leq 12$$

Step-by-Step Solution

Answer

The summed function is graphed by combining ordinates of \(-\frac{1}{4} \cos \left(\frac{\pi}{6} x\right)\) and \(-\frac{1}{2} \cos \left(\frac{\pi}{3} x\right)\) over \(0 \leq x \leq 12\).

Step 1: Identify Summands

The given function is composed of two terms: 1. \(-\frac{1}{4} \cos \left(\frac{\pi}{6} x\right)\) 2. \(-\frac{1}{2} \cos \left(\frac{\pi}{3} x\right)\).We need to understand that these are the individual functions whose ordinates (y-values) we will add together over the interval \(0 \leq x \leq 12\).

Step 2: Graph the First Summand

Graph the function \(y = -\frac{1}{4} \cos \left(\frac{\pi}{6} x\right)\) over the interval \(0 \leq x \leq 12\). The amplitude is \(-\frac{1}{4}\), and the period is determined by \(\frac{2\pi}{\frac{\pi}{6}} = 12\). So this function completes one full cycle from \(x=0\) to \(x=12\).

Step 3: Graph the Second Summand

Graph the function \(y = -\frac{1}{2} \cos \left(\frac{\pi}{3} x\right)\) over the interval \(0 \leq x \leq 12\). The amplitude is \(-\frac{1}{2}\), and the period is determined by \(\frac{2\pi}{\frac{\pi}{3}} = 6\). This means the function completes two full cycles within the interval \(0 \leq x \leq 12\).

Step 4: Add the Ordinates

To find the summed function, at each point \(x\), add the ordinates (y-values) of the two individual functions: \(-\frac{1}{4} \cos \left(\frac{\pi}{6} x\right)\) and \(-\frac{1}{2} \cos \left(\frac{\pi}{3} x\right)\). Thus the summed function is: \[y = -\frac{1}{4} \cos \left(\frac{\pi}{6} x\right) - \frac{1}{2} \cos \left(\frac{\pi}{3} x\right)\].

Step 5: Graph the Summed Function

Plot the summed function \(y = -\frac{1}{4} \cos \left(\frac{\pi}{6} x\right) - \frac{1}{2} \cos \left(\frac{\pi}{3} x\right)\) over the same interval \(0 \leq x \leq 12\) by adding the y-values of the individual graphs at each point \(x\). This will provide a visual representation of the combined effect of both functions on the interval.

Key Concepts

Amplitude

In trigonometry, amplitude measures how far the function's highest point is from its central position. For cosine functions, the amplitude is the absolute value of the coefficient in front of the cosine. In the function \( -\frac{1}{4} \cos\left(\frac{\pi}{6} x\right) \), the amplitude is \( \frac{1}{4} \) and in the function \( -\frac{1}{2} \cos\left(\frac{\pi}{3} x\right) \), the amplitude is \( \frac{1}{2} \). This means:

• The first function peaks at \( \pm\frac{1}{4} \) away from the centerline, starting from negative due to the minus sign.


• The second function peaks at \( \pm\frac{1}{2} \) away from the centerline, also starting from negative because of its minus sign.

Amplitudes help us determine the "height" of our wave, allowing us to predict how far trigonometric functions like cosine will deviate from their median line. As you work with graphing these functions, always pay attention to the amplitude as it affects the wave's peaks and troughs.

Period of Trigonometric Functions

The period of a trigonometric function tells us how long it takes to complete one full cycle. To find the period for a function like \( y = a \cos(bx) \), use the formula \( \frac{2\pi}{b} \).

• For the function \( y = -\frac{1}{4} \cos\left(\frac{\pi}{6} x\right) \), the period is \( \frac{2\pi}{\frac{\pi}{6}} = 12 \). This means it takes \( 12 \) units of \( x \) for the function to complete one cycle.


• For the function \( y = -\frac{1}{2} \cos\left(\frac{\pi}{3} x\right) \), the period is \( \frac{2\pi}{\frac{\pi}{3}} = 6 \). Thus, this function completes one cycle every \( 6 \) units.

The period is crucial because it dictates the frequency of the wave's cycles across a given range. It's what allows us to understand how quickly or slowly a function oscillates within a set interval.

Sum of Ordinates

The sum of ordinates is the process of adding the y-values (ordinates) from two different functions at each point along the x-axis within a defined interval. This results in a new function, which represents the combined effect of the original functions. Let's use the example functions to see how this works:

• The function \( y = -\frac{1}{4} \cos\left(\frac{\pi}{6} x\right) \) is graphed over the interval \( 0 \leq x \leq 12 \).


• Similarly, the function \( y = -\frac{1}{2} \cos\left(\frac{\pi}{3} x\right) \) is also graphed over the same interval.


• At any point \( x \) in our interval, we add the y-values of these two functions to get: \( y = -\frac{1}{4} \cos\left(\frac{\pi}{6} x\right) - \frac{1}{2} \cos\left(\frac{\pi}{3} x\right) \).

The summed function results from this process and its graph provides a visual representation of how combining these two functions affects their overall behavior. Observing the summed function graphically can provide insights into the effects of superposing waveforms in mathematical modeling.