Problem 94 - Precalculus

Question

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

Step-by-Step Solution

Answer

Sum the ordinates of the two cosine functions over the interval to plot their sum: \[-\frac{1}{2} \cos \left(x+\frac{\pi}{3}\right) - 2 \cos \left(x-\frac{\pi}{6}\right), -\pi \leq x \leq \pi \].

Step 1: Understand the Equation

We are given two trigonometric functions, \( y_1 = -\frac{1}{2} \cos \left(x+\frac{\pi}{3}\right) \) and \( y_2 = -2 \cos \left(x-\frac{\pi}{6}\right) \). The task requires finding their sum, \( y = y_1 + y_2 \), over the interval \(-\pi \leq x \leq \pi\).

Step 2: Write the Sum of Functions

We need to write the equation for the summed function: \[ y = -\frac{1}{2} \cos \left(x+\frac{\pi}{3}\right) - 2 \cos \left(x-\frac{\pi}{6}\right) \] This equation represents the addition of the ordinates of the two functions at each point \( x \) in the given interval.

Step 3: Analyze the Functions

Identify characteristics of the cosine functions: - The first function is scaled and inverted by \(-\frac{1}{2}\) and shifted left by \( \frac{\pi}{3} \).- The second function is scaled by \(-2\) and shifted right by \( \frac{\pi}{6} \). These shifts and coefficients affect the wave's period, amplitude, and phase.

Step 4: Choose Key Points in the Interval

To adequately sketch the graph, select key points within the interval \(-\pi \leq x \leq \pi\) such as \(-\pi, -\frac{\pi}{2}, 0, \frac{\pi}{2}, \pi\), etc. These will help in checking the behavior of the summed function at strategic points.

Step 5: Calculate Ordinates at Key Points

Evaluate the sums \( y = y_1 + y_2 \) at each of the key points:- For \( x = -\pi \), compute \(-\frac{1}{2} \cos \left(-\pi+\frac{\pi}{3}\right) - 2 \cos \left(-\pi-\frac{\pi}{6}\right)\).- Repeat for other points to plot how \( y \) behaves across the interval.

Step 6: Plot the Summed Function

Using the calculations from Step 5, plot \( y \) on the graph for each key point within the interval. Sketch the curve that reflects the calculated ordinates, making sure to connect these points smoothly to show periodic behavior, considering both phase shifts and amplitudes.

Key Concepts

Ordinates in Trigonometric Graphing

When you're graphing trigonometric functions, understanding the concept of "ordinates" is essential. Ordinates are simply the y-values of a function. In our context, they are the specific values along the vertical axis that you'll find by substituting different x-values into your function equations.

To sketch graphs of summed functions, as in this exercise, you need to find and "add" these ordinates from each individual function at particular x-values over a specified interval. This process involves evaluating each function separately at a given x-value, and then taking those two results and adding them to get the ordinate of the summed function. This sum is what you'll plot at each key point to construct the graph of the summed function. It's like taking separate snapshots and merging them into one picture that shows the true form of both underlying functions combined.

Summed Functions and Their Graphs

In trigonometric graphing, a "summed function" is created by adding two or more individual functions. This results in a new function that reflects characteristics from each of the original functions while blending them together into a cohesive graph.

For the given problem, you are working with two cosine functions:

• The first: y_1 = -\frac{1}{2} \cos \left(x+\frac{\pi}{3}\right)
• The second: y_2 = -2 \cos \left(x-\frac{\pi}{6}\right)

The summed function is represented by \[ y = y_1 + y_2 \]By calculating the ordinates for each cosine component and then summing them, you can derive a new set of y-values, describing a completely new wave pattern over the interval from -\pi to \pi. This summed function will uniquely reflect combined waves, showcasing new peaks, troughs, and zero crossings where these functions interact.

Understanding Phase Shift in Functions

"Phase Shift" is a core characteristic of trigonometric functions that describes how much a function is shifted horizontally. In the expression inside the cosine function, \(y = -\frac{1}{2} \cos \left(x+\frac{\pi}{3}\right)\), the \(+\frac{\pi}{3}\) indicates a phase shift to the left by \(\frac{\pi}{3}\).

Likewise, in the function \(y = -2 \cos \left(x-\frac{\pi}{6}\right)\), the \(-\frac{\pi}{6}\) indicates a shift to the right by \(\frac{\pi}{6}\).This shifting changes the starting points of the cosine curves on the x-axis, meaning they begin their upward or downward movements from different x-values.

Understanding phase shift helps you predict where each wave starts and how they line up relative to one another. When combined in a summed function, these phase shifts can either amplify or cancel parts of the wave, leading to new patterns.

The Role of Amplitude in Trigonometric Functions

"Amplitude" in trigonometric functions determines the height of the wave from its central axis, dictating how far it swings both above and below this midpoint.

In the function \(y = -\frac{1}{2} \cos \left(x+\frac{\pi}{3}\right)\), the amplitude is \(-\frac{1}{2}\), and for \(y = -2 \cos \left(x-\frac{\pi}{6}\right)\), it is \(-2\). These negative values indicate the waves are inverted.

• An amplitude of \(\frac{1}{2}\) means that the maximum and minimum heights from the central value will be reduced, resulting in less pronounced peaks and valleys.
• An amplitude of \(2\) makes the wave more pronounced, with peaks and troughs that are twice as high and low compared to a standard cosine wave without scaling.

Amplitude controls the extent of these oscillations, shaping how "tall" or "short" the wave appears. As you sum the ordinates, these differing amplitudes blend to form a new wave height pattern, adding to the complexity and richness of the summed function’s graph.