Problem 23 - Precalculus: A Concise Course

Question

Describe the relationship between the graphs of \(f\) and \(g\). Consider amplitude, period, and shifts. $$ \begin{array}{l} f(x)=\cos x \\ g(x)=\cos 2 x \end{array} $$

Step-by-Step Solution

Answer

Both functions have the same amplitude of 1 and no shifts. However, \(g(x) = cos(2x)\) has a period of \(\pi\), which means it oscillates twice as fast as \(f(x) = cos(x)\) which has a period of \(2\pi\).

Step 1: Calculate Amplitude

The amplitude of a function corresponds to the absolute value of the highest vertical distance from the function's midline. For both \(f(x) = cos(x)\) and \(g(x) = cos(2x)\), the amplitudes are 1 because neither function has a coefficient in front of the cosine.

Step 2: Find out the Period

The period of a function is the length of one complete wave cycle. In general, the period of a cosine function is given by \(2\pi /│b│\), where \(b\) is the number multiplying \(x\). For \(f(x) = cos(x)\), \(b\) is 1, so the period is \(2\pi\). For \(g(x) = cos(2x)\), \(b\) is 2, so the period is \(\pi\). Thus, the \(g(x) = cos(2x)\) function completes a wave cycle twice as fast as \(f(x)=cos(x)\).

Step 3: Calculate Shifts

A shift refers to a horizontal or vertical displacement of a function. Normal cosine functions don't have any shifts. In this case, both \(f(x) = cos(x)\) and \(g(x) = cos(2x)\) have no shifts because there are no added or subtracted constants.

Key Concepts

Amplitude

Amplitude refers to the maximum amount a wave goes up or down from a central axis. It represents the "height" of the wave. In trigonometric functions, especially cosine and sine, amplitude is determined by the coefficient in front of the trigonometric function.For example, if you have a function like \(f(x) = a \cdot \cos(bx)\), the amplitude is given by the absolute value of \(a\). This tells us how much the wave stretches vertically.In the functions provided in the exercise:

• \(f(x) = \cos(x)\) has an amplitude of 1, because there is no coefficient in front of \(\cos(x)\) other than the implied 1.
• \(g(x) = \cos(2x)\) also has an amplitude of 1 for the same reason.

The amplitude is crucial because it affects how "tall" or "short" the graph appears, which can change the interpretation of data modeled by the function.

Period

Period refers to the length of one full cycle of the wave before it repeats itself. It's an essential feature of periodic functions like sine and cosine, as it defines how quickly the function oscillates.For cosine functions, the period can be calculated using the formula:\[\text{Period} = \frac{2\pi}{|b|}\]where \(b\) is the coefficient of \(x\) in the function \(f(x) = a \cdot \cos(bx)\). The absolute value ensures we get a positive period.In the exercise:

• For \(f(x) = \cos(x)\), since \(b = 1\), the period is \(2\pi\).
• For \(g(x) = \cos(2x)\), \(b = 2\), so the period is \(\pi\).

This shorter period means \(g(x)\) completes its cycles twice as fast as \(f(x)\). This quick repetition is key when modeling phenomena that change rapidly over time.

Horizontal Shifts

Horizontal shifts, also known as phase shifts, occur when a trigonometric graph is moved left or right along the x-axis. This happens when a constant is added or subtracted inside the function's argument. In mathematical terms, this could look like \(f(x) = \cos(bx + c)\).Generally, the phase shift can be determined by solving:\[\text{Horizontal Shift} = -\frac{c}{b}\]where \(c\) is the constant being added or subtracted.For the functions in the exercise:

• \(f(x) = \cos(x)\) has no horizontal shift since there's no constant added or subtracted inside the cosine function.
• Similarly, \(g(x) = \cos(2x)\) also has no horizontal shift, as its argument is simply \(2x\).

Understanding horizontal shifts is important when aligning trigonometric models with real-world data, such as synchronizing phases in wave patterns or timing signals.