Problem 38 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch the graphs of \(f\) and \(g\) in the same coordinate plane. (Include two full periods.) $$\begin{aligned} &f(x)=2 \cos 2 x\\\ &g(x)=-\cos 4 x \end{aligned}$$

Step-by-Step Solution

Answer

The graphs of \(f(x)\) and \(g(x)\) are periodic graphs with different amplitudes, frequencies and orientations. Function \(f(x) = 2 \cos 2x\) has double amplitude and frequency compared to the standard cosine function while \(g(x) =-\cos 4x\) is a cosine function with a reflection on the x-axis and quadruple frequency.

Step 1: Understand the basic shape

The basic shape of a cosine function is a wave that starts at a peak, drops to a trough and then rises back to a peak. It is periodic, meaning it repeats this pattern indefinitely.

Step 2: Examine function f(x)

For function \(f(x)=2 \cos 2 x\), the coefficient '2' in front of the \(\cos\) function indicates that the amplitude of the function is 2. And the number '2' multiplying \(x\) inside the \(\cos\) function indicates that the frequency of the function is 2. So, expect two full waves within the range \(0 \leq x \leq 2\pi\). Draw these two waves with the amplitude of 2.

Step 3: Examine function g(x)

Now for function \(g(x)=-\cos 4 x\), the negative sign in front of the \(\cos\) function indicates that the function is reflected at the x-axis. This means that instead of starting at a peak, it starts at a trough. The number '4' multiplying \(x\) inside the \(\cos\) function indicates that the frequency of the function is 4. So, expect four full waves within the range \(0 \leq x \leq 2\pi\). Draw these four waves, remembering to start with a trough due to the negative sign.

Step 4: Combine the sketches

Using the same axes, plot the sketches of \(f(x)\) and \(g(x)\) together. Note that they will intersect at several points, but the shape, frequency, and amplitude of the graphs are different based on their individual functions.

Key Concepts

Amplitude of Cosine Function

In trigonometry, the amplitude of a cosine function is a measure of how far the peaks and troughs of the wave are from its central axis; essentially, it represents the height of the wave. For any cosine function in the form of f(x) = A cos(Bx+C) + D, the absolute value of A determines the amplitude. For the function f(x) = 2 cos(2x), the amplitude is the coefficient in front of the cosine, which is 2. This means the graph will reach up to 2 units above the central axis and 2 units below, creating a wave with a total vertical span of 4 units.

Understanding amplitude is crucial for graphing as it allows you to accurately represent the vertical stretching or compressing of the graph. When the amplitude is greater than 1, the wave is taller; when it's less than 1, the graph appears flatter.

Frequency of Cosine Function

The frequency of a cosine function reflects how often the waves repeat themselves over a certain distance, usually over 2π for trigonometric functions. It's related to the coefficient B in the standard form of the function f(x) = A cos(Bx+C) + D. For the function g(x) = -cos(4x), the frequency is determined by the number 4, which multiplies the variable x.

In graphing cosine functions, higher frequencies mean that the waves are closer together, or that there are more waves per unit length. In the case of \(g(x) = -cos(4x)\), you would graph 4 complete waves within the interval from 0 to 2π. Frequency is essential for understanding how 'tight' or 'spread out' the waves of the function are.

Period of Trigonometric Function

The period of a trigonometric function is the horizontal length of one complete wave cycle. For a standard cosine function y = cos(x), the period is 2π, since this is the distance over which the wave repeats. However, when a coefficient multiplies the variable x, this changes. The formula for the period of a cosine or sine function is P = (2π)/|B|, where B is the coefficient from f(x) = A cos(Bx+C) + D.

Applying this to the functions from the exercise, for f(x) = 2 cos(2x), the period is P = (2π)/|2| = π, which means one full wave is completed every π units along the x-axis. The concept of period helps in plotting the wave by informing us about the intervals where the cycle starts and ends.

Phase Shift in Trigonometry

A phase shift in trigonometry is a horizontal shift of the graph either to the right or left. The shift happens when a constant C is added to or subtracted from the variable x in the function's standard form of f(x) = A cos(Bx+C) + D. A positive C indicates a shift to the left, while a negative C indicates a rightward shift. The amount of the shift is given by |C|/B.

In the given functions f(x) = 2 cos(2x) and g(x) = -cos(4x), there is no phase shift since there is no added or subtracted constant with the variable x. To introduce a phase shift, you could alter the functions, for example, h(x) = cos(x - π/2) would shift the function to the right by π/2 units.