Find Amplitude, Period, and Phase Shift y=-2cos(x+pi/6)

Solution:

Step 1: Identify the coefficients for amplitude, period, phase shift, and vertical shift

Given the function $y=-2\cos (x+\frac{\pi }{6})$, we can determine:

• Amplitude coefficient ($a$): $-2$
• Period coefficient ($b$): $1$
• Phase shift coefficient ($c$): $\frac{\pi }{6}$ (Note the negative sign inside the cosine function)
• Vertical shift ($d$): $0$

Step 2: Calculate the amplitude

The amplitude is the absolute value of $a$.

Amplitude: $|-2|=2$

Step 3: Determine the period of the cosine function

Step 3.1

The period is found using the formula $\frac{2\pi }{|b|}$.

Step 3.2

Substitute $b=1$ into the period formula.

Period: $\frac{2\pi }{|1|}$

Step 3.3

Since the absolute value of $1$ is $1$, we have:

Period: $\frac{2\pi }{1}$

Step 3.4

Calculating the period gives us $2\pi$.

Period: $2\pi$

Step 4: Compute the phase shift

Step 4.1

The phase shift is given by $\frac{c}{b}$.

Step 4.2

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

Phase Shift: $\frac{-\frac{\pi }{6}}{1}$

Step 4.3

Simplifying the fraction gives us the phase shift.

Phase Shift: $-\frac{\pi }{6}$

Step 5: Summarize the properties of the cosine function

• Amplitude: $2$
• Period: $2\pi$
• Phase Shift: $-\frac{\pi }{6}$ (which is a shift to the left by $\frac{\pi }{6}$)
• Vertical Shift: None (since $d=0$)

Step 6: Conclusion

The amplitude, period, and phase shift of the function $y=-2\cos (x+\frac{\pi }{6})$ are $2$, $2\pi$, and $-\frac{\pi }{6}$, respectively, with no vertical shift.

Knowledge Notes:

To analyze the function $y=a\cos (bx-c)+d$ and find its amplitude, period, phase shift, and vertical shift, we use the following knowledge:

1. Amplitude: The amplitude of a trigonometric function is the absolute value of the coefficient $a$. It represents the maximum value the function reaches above or below its midline, which is the vertical shift $d$.

2. Period: The period of a trigonometric function is the length of one complete cycle of the function. For a cosine function, the period is calculated as $\frac{2\pi }{|b|}$, where $b$ is the coefficient in front of $x$.

3. Phase Shift: The phase shift is the horizontal shift of the function along the x-axis. It is calculated as $\frac{c}{b}$, where $c$ is the constant added or subtracted inside the function's argument. A positive phase shift indicates a shift to the right, while a negative phase shift indicates a shift to the left.

4. Vertical Shift: The vertical shift is the value $d$, which moves the function up or down along the y-axis.

5. Absolute Value: The absolute value of a number is its distance from zero on the number line, regardless of direction. It is always non-negative.

By applying these concepts, we can analyze the given trigonometric function and determine its key characteristics.