Problem 61 - Precalculus: Functions and Graphs

Question

Sketch the graph of the equation. $$y=x+\cos x$$

Step-by-Step Solution

Answer

The graph is a wave pattern oscillating around the line \( y = x \).

Step 1: Understand the Components

To sketch the graph of the equation \( y = x + \cos x \), first recognize the components involved. The graph consists of two parts: a linear part \( y = x \) and a trigonometric part \( y = \cos x \).

Step 2: Sketch the Basic Functions Separately

First, sketch \( y = x \), which is a straight line passing through the origin with a slope of 1. Next, sketch \( y = \cos x \), which is a cosine wave with an amplitude of 1, fluctuating above and below the x-axis with a period of \( 2\pi \).

Step 3: Analyze the Combination

Consider that the cosine wave, \( \cos x \), modulates the linear function \( y = x \). This means that for every value of x, the graph of \( y = x + \cos x \) will be the line \( y = x \) shifted up or down by the cosine value. This creates a wave-like pattern centered on the line \( y = x \).

Step 4: Identify Key Points

Identify key points to help plot the graph. For example, at \( x = 0 \), \( y = 0 + \cos(0) = 1 \). At \( x = \pi \), \( y = \pi - 1 \), and at \( x = 2\pi \), \( y = 2\pi + 1 \). These points help to sketch the fluctuating pattern.

Step 5: Plot and Connect Points

Plot the identified key points on a coordinate plane. Use the wave pattern of the cosine function to connect the points, ensuring that the wave fluctuates around the line \( y = x \) according to the values of \( \cos x \). The graph should resemble a series of upward waves along the line \( y = x \).

Step 6: Add Labels and Axis

Ensure to label the x and y axes. Mark the key intersections and turning points, such as the maximum and minimum values caused by the cosine fluctuation. These details will help in better understanding the graph.

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. They are fundamental in studying wave patterns and oscillations. The primary trigonometric functions are sine (\(\sin x\)), cosine (\(\cos x\)), and tangent (\(\tan x\)). These functions are periodic, meaning they repeat at regular intervals.
In the context of the equation \( y = x + \cos x\), the cosine function introduces a wave-like oscillation to the linear function \(y = x\). The cosine wave adds undulating peaks and troughs to the straightforward line. By understanding trigonometric functions, you can greatly appreciate the regular and smooth fluctuations they bring.

Linear Functions

Linear functions come in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. In simpler terms, these are functions that produce straight lines when graphed.
With \(y = x\), our linear function is the simplest: it has a slope of 1 and a y-intercept of 0. This means the line rises consistently, with each step to the right up the same amount vertically. When combined with other function types, like the trigonometric cosine, it acts as the baseline that the other function modifies.

• Easy to understand and plot
• Represents constant straight movement
• Acts as a reference for understanding more complex graphs

Amplitude

Amplitude describes how far the graph of a function stretches from its central axis, representing the maximum deviation of the function from its average position. In simpler terms, it's the height of the wave. For the cosine function \(\cos x\), the amplitude is 1, which indicates the highest and lowest points the curve reaches.
In the \(y = x + \cos x\) equation, the amplitude of the cosine function affects how much the combined graph oscillates around the line \(y = x\). The wave-like pattern affects how prominently it humps above and dips below the linear path, mirroring the cosine's intrinsic oscillating nature.

Periodicity

Periodicity refers to the repeating nature of a function over regular intervals. In trigonometric contexts, periodicity is a fundamental aspect that you observe as a pattern consistently repeats after a certain interval.
The cosine function \(\cos x\) features a period of \(2\pi\). This means every \(2\pi\) units along the x-axis, the cosine curve completes a cycle and starts to repeat itself. In our exercise, this periodic nature means that while \(y = x\) is a continuously increasing line, the function \(y = x + \cos x\) oscillates repeatedly due to cosine's periodicity. This results in the graph having a rhythmic pattern of peaks and troughs as it follows the line.