Find Amplitude, Period, and Phase Shift y=7cos(3x)

Solution:

Step 1: Identify the coefficients in the standard cosine function format

The given function $y=7\cos (3x)$ can be compared to the standard form $a\cos (bx-c)+d$ to determine the amplitude, period, phase shift, and vertical shift.

• Amplitude coefficient ($a$): $7$
• Frequency coefficient ($b$): $3$
• Phase shift coefficient ($c$): $0$
• Vertical shift ($d$): $0$

Step 2: Calculate the amplitude

The amplitude is the absolute value of the amplitude coefficient ($a$).

Amplitude: $|7|=7$

Step 3: Determine the period of the function

Step 3.1

The period $T$ of a cosine function is found using the formula $T=\frac{2\pi }{|b|}$.

Step 3.2

Substitute the frequency coefficient $b=3$ into the period formula.

Period: $T=\frac{2\pi }{|3|}$

Step 3.3

Since the absolute value of $3$ is $3$, the period is $T=\frac{2\pi }{3}$.

Step 4: Compute the phase shift

Step 4.1

The phase shift is calculated using the formula $\frac{c}{b}$.

Step 4.2

Insert the values for $c=0$ and $b=3$ into the phase shift formula.

Phase Shift: $\frac{0}{3}$

Step 4.3

Dividing $0$ by $3$ yields a phase shift of $0$.

Step 5: Summarize the trigonometric function's characteristics

• Amplitude: $7$
• Period: $\frac{2\pi }{3}$
• Phase Shift: $0$ (None)
• Vertical Shift: $0$ (None)

Knowledge Notes:

The amplitude, period, phase shift, and vertical shift are properties of trigonometric functions that describe their shape and position on a graph.

• Amplitude: The amplitude of a trigonometric function is the maximum distance from the midline of the wave to its peak or trough. It is the absolute value of the coefficient $a$ in the standard form $a\cos (bx-c)+d$ or $a\sin (bx-c)+d$.

• Period: The period of a trigonometric function is the length of one complete cycle of the wave. It is determined by the coefficient $b$ and can be calculated using the formula $T=\frac{2\pi }{|b|}$ for cosine and sine functions.

• Phase Shift: The phase shift of a trigonometric function is the horizontal displacement of the function's graph from its usual position. It is calculated using the formula $\frac{c}{b}$, where $c$ is the phase shift coefficient in the standard form. A positive phase shift moves the graph to the right, while a negative phase shift moves it to the left.

• Vertical Shift: The vertical shift is the upward or downward displacement of the graph of the function. It corresponds to the value of $d$ in the standard form. A positive vertical shift moves the graph upwards, and a negative vertical shift moves it downwards.

In the given problem, the trigonometric function $y=7\cos (3x)$ has an amplitude of $7$, a period of $\frac{2\pi }{3}$, no phase shift, and no vertical shift. These properties define the function's graph and can be used to sketch it accurately.