Find Amplitude, Period, and Phase Shift y=2cos(pix-1)+2

Solution:

Step 1:

Identify the coefficients $a$, $b$, $c$, and $d$ from the standard trigonometric form $a\cos (bx-c)+d$ to determine amplitude, period, phase shift, and vertical shift.

• Amplitude coefficient ($a$): $2$
• Frequency coefficient ($b$): $\pi$
• Phase shift coefficient ($c$): $1$
• Vertical shift ($d$): $2$

Step 2:

Calculate the amplitude by taking the absolute value of $a$.

• Amplitude: $\left|2\right|=2$

Step 3:

Determine the period using the formula $\frac{2\pi }{\left|b\right|}$.

Step 3.1:

Compute the period for $2\cos (\pi x-1)$.

Step 3.1.1:

Apply the period formula $\frac{2\pi }{\left|b\right|}$.

Step 3.1.2:

Substitute $\pi$ for $b$ to find the period: $\frac{2\pi }{\left|\pi \right|}$.

Step 3.1.3:

Since $\pi \approx 3.14159265$ is positive, the absolute value is not needed: $\frac{2\pi }{\pi }$.

Step 3.1.4:

Eliminate the common $\pi$ factor.

Step 3.1.4.1:

Simplify by canceling out $\pi$: $\frac{2\cancel{\pi }}{\cancel{\pi }}$.

Step 3.1.4.2:

Finalize the calculation: $2$.

Step 3.2:

Confirm the period is $2$.

Step 3.2.1:

Reapply the period formula $\frac{2\pi }{\left|b\right|}$.

Step 3.2.2:

Insert $\pi$ for $b$ again: $\frac{2\pi }{\left|\pi \right|}$.

Step 3.2.3:

Remove the absolute value as before: $\frac{2\pi }{\pi }$.

Step 3.2.4:

Proceed to cancel the common $\pi$ factor.

Step 3.2.4.1:

Cancel $\pi$: $\frac{2\cancel{\pi }}{\cancel{\pi }}$.

Step 3.2.4.2:

Complete the division: $2$.

Step 3.3:

The period for a sum or difference of trigonometric functions is the maximum of the individual periods, which is $2$.

Step 4:

Calculate the phase shift with the formula $\frac{c}{b}$.

Step 4.1:

The phase shift is derived from $\frac{c}{b}$.

Step 4.2:

Input the known values of $c$ and $b$: Phase Shift: $\frac{1}{\pi }$.

Step 5:

Compile the characteristics of the trigonometric function.

• Amplitude: $2$
• Period: $2$
• Phase Shift: $\frac{1}{\pi }$ (shifted $\frac{1}{\pi }$ units to the right)
• Vertical Shift: $2$

Knowledge Notes:

To analyze and describe the properties of a trigonometric function, particularly a cosine function of the form $y=a\cos (bx-c)+d$, we use the following key concepts:

1. Amplitude: The amplitude of a trigonometric function is the absolute value of the coefficient $a$. It determines the height of the wave's peaks and the depth of its troughs from the function's midline. In the given function, the amplitude is $|a|$.

2. Period: The period of a trigonometric function is the length of one complete cycle of the wave. It is calculated using the formula $\frac{2\pi }{|b|}$, where $b$ is the frequency coefficient. The period tells us how long it takes for the function to repeat its pattern.

3. Phase Shift: The phase shift of a trigonometric function is the horizontal shift along the x-axis. It is determined by the formula $\frac{c}{b}$, where $c$ is the horizontal shift coefficient. A positive phase shift means the function is shifted to the right, while a negative shift indicates a shift to the left.

4. Vertical Shift: The vertical shift is the value $d$ in the function, which moves the function up or down along the y-axis. It changes the midline of the wave but does not affect the amplitude or period.

5. Frequency: The frequency is related to the period and is the number of cycles the function completes in a unit interval. It is given by the reciprocal of the period.

Understanding these properties allows us to graph the trigonometric function accurately and predict its behavior over any interval.