Problem 34 - Algebra and Trigonometry

Question

Graph each function by adding y-coordinates. $$ g(x)=\cos (2 x)+\cos x $$

Step-by-Step Solution

Answer

Add the y-values of \text{{cos}}(\(2x\)) and \text{{cos}}(\(x\)) at chosen x-points and plot the summed points.

Step 1: Understand the Components of the Function

The given function is a combination of two cosine functions: \(g(x)=\text{cos}(2x) + \text{cos}(x)\). To graph this function, start by understanding how the individual \text{cos}(2x) and \text{cos}(x) functions behave. The standard cosine function \text{cos}(x) oscillates between -1 and 1 with a period of \(2\text{{pi}}\). The function \text{cos}(2x) also oscillates between -1 and 1 but with a period of \(\text{{pi}}\), making it oscillate faster.

Step 2: Choose Key Points

Select key points to evaluate the function, usually within one period of \(2\text{{pi}}\). For instance, choose \(x=0, \frac{\text{{pi}}}{4}, \frac{\text{{pi}}}{2}, \frac{3\text{{pi}}}{4}, \text{{pi}}, \frac{5}{4}\text{{pi}}, \frac{3}{2}\text{{pi}}, \frac{7}{4}\text{{pi}}, 2\text{{pi}}\).

Step 3: Calculate y-values for Individual Functions

Calculate the \(y\)-values for \text{cos}(x) and \text{cos}(2x) at the chosen \(x\) points. For example, \(x=0: \text{{cos}}(0) = 1\) and \text{{cos}}(2 \times 0) = 1\( \)x=\frac{\text{{pi}}}{4}: \text{{cos}}(\frac{\text{{pi}}}{4}) \text{\textapprox} 0.707\(, \text{{cos}}(2 \times \frac{\text{{pi}}}{4}) \text{\textapprox} 0\).... Continue for all \(x\) values.

Step 4: Add Function Values

For each \(x\) point, add the corresponding \(y\) values of \text{{cos}}(\(2x\)) and \text{{cos}}(\(x\)). For example, \(g(0) = 1 + 1 = 2\) \(g(\frac{\text{{pi}}}{4}) \text{\textapprox} 0.707 + 0 = 0.707\) .... Continue for all \(x\) values.

Step 5: Plot the Points

On a coordinate plane, plot the points \((x, g(x))\) obtained from the previous step. These points will represent the function \(g(x)=\text{cos}(2x) + \text{cos}(x)\).

Step 6: Draw the Graph

Connect the plotted points smoothly to form the graph of the function \(g(x)=\text{cos}(2x) + \text{cos}(x)\). Ensure the curve follows the oscillatory nature indicated by the cosine components.

Key Concepts

cosine function

To graph the given function, we need to first understand the individual cosine functions involved. The cosine function, denoted as \( \text{cos}(x) \), is a fundamental trigonometric function that oscillates between -1 and 1. It has a smooth, wave-like shape and completes one full cycle every \( 2\text{{π}} \). This means its period, or the horizontal length of one complete wave, is \( 2\text{{π}} \). This function is particularly useful for modeling repetitive patterns, such as sound waves or seasonal changes.

• At \( x = 0 \), \( \text{cos}(0) = 1 \)
• At \( x = \text{π} \), \( \text{cos}(\text{π}) = -1 \)
• At \( x = 2\text{π} \), \( \text{cos}(2\text{π}) = 1 \)

The function \( \text{cos}(2x) \) modifies \( \text{cos}(x) \) by doubling the input value of \( x \). This results in the function oscillating faster, effectively halving its period to \( \text{π} \). For instance, \( \text{cos}(2\text{π}) \) goes through two full cycles within the same \( 2\text{π} \) range.

periodic functions

Periodic functions are those that repeat their values in regular intervals or periods. The cosine function is a perfect example of a periodic function because it repeats its pattern every \( 2\text{π} \). Functions like \( \text{cos}(2x) \) also fall under this category but with a different period, specifically, it repeats every \( \text{π} \).
Keep these points in mind about periodic functions:

• Period: The length of one complete cycle. For \( \text{cos}(x) \), it's \( 2\text{π} \), for \( \text{cos}(2x) \), it's \( \text{π} \).
• Amplitude: The peak value of the function. Cosine functions typically oscillate between -1 and 1, so their amplitude is 1.
• Phase Shift: A horizontal shift in the graph. Although not present in this exercise, phase shift is crucial in other trigonometric functions.

Understanding the periodic nature of these functions helps in predicting their behavior over intervals, making it easier to graph them. The key takeaway is that the initial function and its transformed versions (like \( \text{cos}(2x) \)) give us the integral components we need to create more complex graphs.

function addition

When dealing with functions like \( g(x)=\text{cos}(2x) + \text{cos}(x) \), the principle of function addition plays a crucial role. Adding two functions means adding their y-coordinates at specific x-values.
Here's how you can tackle function addition step-by-step:

• Select key points within one period. For example, for \( x \) values ranging from 0 to \( 2\text{π} \).
• Calculate y-values for each function individually at these points. For \( x=0 \), both \( \text{cos}(0) \) and \( \text{cos}(2 \times 0) \) are 1.
• Add these y-values together to get the resultant function's y-values. For \( x=0 \), this sum would be \( 1+1=2 \).
• Plot these combined y-values against their corresponding x-values.

Function addition essentially merges two or more functions into one, creating a new wave pattern that reflects the combined amplitude and phase shifts of the individual functions. Understanding this concept is key to mastering the graphing of compound trigonometric functions.