Problem 26 - Algebra and Trigonometry Real Mathematics, Real People

Question

Describe the relationship between the graphs of \(f\) and \(g .\) Consider amplitudes, periods, and shifts. $$\begin{aligned} &f(x)=\cos 4 x\\\ &g(x)=-2+\cos 4 x \end{aligned}$$

Step-by-Step Solution

Answer

Both \(f(x)\) and \(g(x)\) have the same amplitude of 1 and the same period of \(\frac{2\pi}{4}\). However, \(g(x)\) is \(f(x)\) shifted downward by 2 units.

Step 1: Identify the Period and Amplitude of Each Function

The period of both functions is the same, it's \(\frac{2\pi}{4}\), because the coefficient of \(x\) in the cosine function determines the period. The amplitude of \(f(x)\) is 1 (the absolute value of the coefficient in front of the cosine), and the amplitude of \(g(x)\) is also 1.

Step 2: Identify the Shifts

Function \(f(x)\) has no shift because there are no numbers added or subtracted directly to the cosine function. However, function \(g(x)\) is shifted down by 2 units, evidenced by the -2 added to the cosine function.

Step 3: Describing the Relationship

The amplitude of both functions is the same (i.e., 1), and they have the same period. However, \(g(x)\) is \(f(x)\) shifted down by 2 units.

Key Concepts

Amplitude of Trigonometric Functions

Imagine a calm sea with waves gently rising and falling. The amplitude in trigonometry is like the height of those waves. It measures how far the peaks and troughs of a trigonometric graph are from the central line, or the 'equilibrium'.

Specifically, for the function
\( f(x) = \text{cos} (ax) \), where \( a \) is a real number, the amplitude is the absolute value of the coefficient in front of the cosine function. In the case of our exercise's function
\( f(x) = \text{cos} (4x) \), the coefficient is 1, indicating an amplitude of 1. This means the graph oscillates 1 unit above and below the line \( y = 0 \).

No matter how the graph stretches or shrinks horizontally (which we'll discuss with the period), the amplitude remains constant, unless there's a vertical stretch factor—a coefficient placed directly in front of the trigonometric function that would amplify the 'wave heights', so to speak.

Period of Trigonometric Functions

The period of a trigonometric function is akin to the length of one complete cycle of its graph. Take a round trip on a Ferris wheel as an analogy—starting from one point and coming back around to it is one period. For cosine and sine functions, the standard period is
\( 2\text{π} \), but this period can be altered by a coefficient attached to the variable inside the function.

For instance,
\( f(x) = \text{cos} (bx) \) where \( b \) affects the period, making it
\( \frac{2\text{π}}{|b|} \). In our exercise, the period of both
\( f(x) \) and
\( g(x) \) is
\( \frac{2\text{π}}{4} = \frac{\text{π}}{2} \), since the coefficient \( b \) is 4. This shortened period means our 'Ferris wheel' goes around much faster, completing a cycle in just a quarter of the time it usually takes.

Vertical Shifts in Trigonometric Graphs

Now, picture you're on a swing—any movement up or down from the resting position is a vertical shift. In trigonometry, a vertical shift moves the entire graph of a function up or down on the coordinate plane.

When a constant is added or subtracted from the trigonometric function
\( f(x) = \text{cos} (x) \), as in
\( g(x) = -2 + \text{cos} (4x) \) from our exercise, the graph is translated vertically. This -2 in
\( g(x) \) means the graph is shifted down by 2 units. It's essential to note that vertical shifts do not affect the amplitude or period; think of it as sliding the whole swing set lower or higher, without changing the swing's range of motion. Thus,
\( g(x) \) experiences a downward shift, creating a visual 'offset' when compared to
\( f(x) \) without altering the swing's amplitude or the speed at which it moves (period).