Problem 72 - Precalculus Essentials

Question

Graph \(f, g,\) and \(h\) in the same rectangular coordinate system for \(0 \leq x \leq 2 \pi .\) Obtain the graph of h by adding or subtracting the corresponding \(y\) -coordinates on the graphs of \(f\) and \(g\) $$f(x)=2 \cos x, g(x)=\cos 2 x, h(x)=(f+g)(x)$$

Step-by-Step Solution

Answer

The graph of h(x) is obtained by adding the corresponding y-coordinates of the graphs of f(x) = 2cosx and g(x) = cos2x, within the range 0 ≤ x ≤ 2π.

Step 1: Understanding Cosine Functions

Cosine functions have a period of \(2\pi\), and their values range between -1 and 1. This information will be useful in graphing functions f(x) and g(x). The function f(x) = 2cosx is a vertically scaled version of the standard cosine function, while g(x) = cos2x is horizontally compressed.

Step 2: Graphing f(x) and g(x)

On a graph, start by marking the x-axis from 0 to \(2\pi\). For f(x), plot the key points: at x=0, \(f(x) = 2\); at x=\(\pi/2\), \(f(x) = 0\); at x=\(\pi\), \(f(x) = -2\); at x=\(3\pi/2\), \(f(x) = 0\); at x=\(2\pi\), \(f(x) = 2\). For g(x), plot the key points: at x=0, \(g(x) = 1\); at x=\(\pi/4\), \(g(x) = 0\); at x=\(\pi/2\), \(g(x) = -1\); at x=\(3\pi/4\), \(g(x) = 0\); at x=\(\pi\), \(g(x) = 1\); continue similarly until \(x=2\pi\). Connect these points smoothly to create the graphs.

Step 3: Creating the Graph of h(x)

To create the plot for h(x), add the corresponding y-values of f(x) and g(x) at each point in the x-range. i.e., for each x, calculate h(x) as f(x) + g(x). Plot these points on the graph and join them smoothly.

Key Concepts

Cosine Function Properties

Cosine functions, which are fundamental in the study of trigonometry, showcase specific properties that make them predictable and useful in various applications.

Firstly, the cosine graph exhibits a wave-like pattern that repeats after every period of \(2\pi\) radians. This repeating segment is known as the cycle of the function. Any cosine function will exhibit a maximum value of 1 and a minimum value of -1 within one cycle, which corresponds to the peak and the trough of its wave, respectively.

Secondly, its symmetry stands out. Cosine functions are even, which means they have an axis of symmetry along the y-axis. This explains why the cosine graph mirrors itself on either side of the y-axis.

Additionally, the initial phase of a cosine function starts at a maximum point when considered in its basic form, \(\cos(x)\), without any transformations. The cosine of zero is 1, with the graph descending from this peak as the angle increases from zero radians.

Amplitude and Period of Cosine Functions

In the realm of trigonometry, amplitude and period are characteristics of cosine functions that determine their stretch and cycle duration.

The

Amplitude

refers to the vertical stretch of the function. In essence, it's the distance from the middle of the wave (the horizontal axis) to its peak (maximum) or trough (minimum). The standard cosine function has an amplitude of 1, but this can be modified by a coefficient in front of the function. For instance, in the function \(f(x)=2\cos(x)\), the amplitude is 2, indicating that the wave's peak is now at 2 instead of 1.

The

Period

describes how long it takes for the cosine function to complete one full cycle. The standard period of a cosine function is \(2\pi\) radians. However, alterations to the function's frequency can change its period. In the function \(g(x)=\cos(2x)\), the period is reduced to \(\pi\) because the cosine function now cycles twice as fast due to the horizontal compression by a factor of 2.

Transformations of Cosine Functions

Transformations play a pivotal role in altering the graphical representation of cosine functions, which can change their amplitude, period, phase shift, and vertical shift.

Vertical Scaling

: When a cosine function is multiplied by a factor, it affects the amplitude. As seen with \(f(x)=2\cos(x)\), the factor 2 amplifies the wave height without changing the period.

Horizontal Scaling

: When the argument of the cosine function is multiplied by a factor, as in \(g(x)=\cos(2x)\), it compresses or stretches the function horizontally, affecting the period. A factor greater than 1 will compress the function, reducing its period, while a factor less than 1 will stretch it, increasing the period.

Phase Shifts

: These occur when the entire graph is moved horizontally. They are caused by adding or subtracting a constant inside the function’s argument. A positive value shifts the graph to the right, while a negative one shifts it to the left.

Vertical Shifts

: Altering the baseline of a cosine function involves adding or subtracting a value outside of the function itself, which shifts the graph up or down accordingly.