Problem 3 - Precalculus: Functions and Graphs

Question

Find the amplitude and the period and sketch the graph of the equation: A. \(y=3 \cos x\) B. \(y=\cos 3 x\) C. \(y=\frac{1}{3} \cos x\) D. \(y=\cos \frac{1}{3} x\) E. \(y=2 \cos \frac{1}{3} x\) F. \(y=\frac{1}{2} \cos 3 x\) G. \(y=-3 \cos x\) H. \(y=\cos (-3 x)\)

Step-by-Step Solution

Answer

Amplitude and period are: A: 3, \(2\pi\); B: 1, \(\frac{2\pi}{3}\); C: \(\frac{1}{3}\), \(2\pi\); D: 1, \(6\pi\); E: 2, \(6\pi\); F: \(\frac{1}{2}\), \(\frac{2\pi}{3}\); G: 3, \(2\pi\); H: 1, \(\frac{2\pi}{3}\).

Step 1: Identify the standard cosine function properties

The standard cosine function is given by \(y = \cos x\), where the amplitude is 1 and the period is \(2\pi\). We'll use this information to determine the amplitude and period of each function listed in the exercise.

Step 2: Calculate amplitude and period for A. \(y = 3 \cos x\)

The amplitude is the coefficient in front of the cosine, which is 3. The period remains \(2\pi\) since there is no horizontal stretching or compressing factor inside the cosine function.

Step 3: Calculate amplitude and period for B. \(y = \cos 3x\)

The amplitude is 1 since there is no coefficient in front of the cosine. The period is \(\frac{2\pi}{3}\) because the function is \(\cos (bx)\) with \(b=3\), and the period formula is \(\frac{2\pi}{b}\).

Step 4: Calculate amplitude and period for C. \(y = \frac{1}{3} \cos x\)

The amplitude is \(\frac{1}{3}\). The period is \(2\pi\), as the coefficient of \(x\) inside \(\cos\) is 1, resulting in no change in the period.

Step 5: Calculate amplitude and period for D. \(y = \cos \frac{1}{3} x\)

The amplitude is 1, as there is no coefficient in front of \(\cos\). The period is \(6\pi\) since the coefficient \(b\) is \(\frac{1}{3}\) and the period is given by \(\frac{2\pi}{b}\).

Step 6: Calculate amplitude and period for E. \(y = 2 \cos \frac{1}{3} x\)

The amplitude is 2. The period is \(6\pi\), calculated from \(\frac{2\pi}{b}\) with \(b = \frac{1}{3}\).

Step 7: Calculate amplitude and period for F. \(y = \frac{1}{2} \cos 3x\)

The amplitude is \(\frac{1}{2}\). The period is \(\frac{2\pi}{3}\).

Step 8: Calculate amplitude and period for G. \(y = -3 \cos x\)

The amplitude is the absolute value of the coefficient, which is 3. The period is \(2\pi\), as the negative sign only affects the direction, not the period.

Step 9: Calculate amplitude and period for H. \(y = \cos (-3x)\)

The amplitude is 1, and the period remains \(\frac{2\pi}{3}\). The negative sign inside the cosine argument only reflects the graph across the y-axis, not affecting the period.

Key Concepts

Cosine Function Properties

The cosine function is an essential trigonometric function characterized by a wave-like pattern. At its core, the standard cosine function is expressed mathematically as \(y = \cos x\). Here are some key properties:

• **Amplitude:** The amplitude is the maximum distance of the wave from the central axis, essentially the "height" of the wave. In \(y = \cos x\), the amplitude is 1.
• **Period:** The period is the distance the function travels along the x-axis before repeating. For \(y = \cos x\), it is \(2\pi\).
• **Symmetry:** The cosine function is even, meaning \(\cos(-x) = \cos x\), keeping it symmetrical about the y-axis.

By understanding these properties, students can predict how changes to coefficients and arguments in a cosine function affect its graph and characteristics. For instance, altering the coefficient in front of \(\cos x\) changes the amplitude, and modifying arguments inside \(\cos x\) itself affects the period.

Graphing Trigonometric Functions

Graphing trigonometric functions like the cosine function starts with identifying its basic properties—amplitude, period, phase shift, and vertical shift. The graph of \(y = \cos x\) is a continuous wave that repeats every \(2\pi\). Here's how we can graph it:

• **Begin with the basic curve**: Start plotting from point (0,1), moving through key points like \((\pi/2, 0)\), \((\pi, -1)\), \((3\pi/2, 0)\), and back to \((2\pi, 1)\).
• **Amplitude Scaling**: If there is a coefficient, like in \(y = 3\cos x\), multiply the y-values by the amplitude (3 in this case).
• **Period Adjustment**: Changing the coefficient of \(x\) affects the period. For instance, \(y = \cos 3x\) compresses the period to \(\frac{2\pi}{3}\). Calculate this new period and adjust your x-axis scale accordingly.

Graphing becomes a straightforward task of applying these transformations to the function's standard form. Practice makes these steps instinctive, allowing students to predict and draw the shape of these functions with ease.

Amplitude and Period Calculation

To calculate the amplitude and period of trigonometric functions, such as those in the given exercise, follow these steps:

• **Amplitude Calculation**: Identify the coefficient in front of the cosine (or sine). The amplitude is the absolute value of this coefficient. For example, in \(y = -3 \cos x\), the amplitude is \(|-3| = 3\).
• **Period Calculation**: Look inside the argument of the cosine or sine function. If it's \(\cos(bx)\), the period is given by \(\frac{2\pi}{b}\). Therefore, \(y = \cos 3x\) has a period of \(\frac{2\pi}{3}\).

For both amplitude and period, remember:- **Amplitude** impacts the range of vertical stretch or shrink.- **Period** influences how wide or narrow the wave extends horizontally.These calculations allow you to alter and understand the wave without directly plotting, making it easier to sketch or conceive of transformations to the graph.