Find Amplitude, Period, and Phase Shift y=5tan(x-pi/3)

Solution:

Step 1:

Identify the coefficients and constants in the expression $5\tan (bx-c)+d$ to determine the amplitude, period, phase shift, and vertical shift.

• Amplitude coefficient $a=5$
• Period coefficient $b=1$
• Phase shift constant $c=\frac{\pi }{3}$
• Vertical shift constant $d=0$

Step 2:

Recognize that the tangent function does not have an amplitude due to its infinite range.

• Amplitude: Not applicable

Step 3:

Calculate the period of the tangent function $5\tan (x-\frac{\pi }{3})$.

• The period formula is $\frac{\pi }{|b|}$.

• Substitute $b=1$ into the period formula.

• The absolute value of $b$ is $1$.

• The period is $\frac{\pi }{1}$, which simplifies to $\pi$.

Step 4:

Determine the phase shift using the formula $\frac{c}{b}$.

• The phase shift formula is $\frac{c}{b}$.

• Insert $c=\frac{\pi }{3}$ and $b=1$ into the phase shift formula.

• The phase shift is $\frac{\frac{\pi }{3}}{1}$, which simplifies to $\frac{\pi }{3}$.

Step 5:

Summarize the characteristics of the trigonometric function.

• Amplitude: Not applicable
• Period: $\pi$
• Phase Shift: $\frac{\pi }{3}$ (shifted to the right by $\frac{\pi }{3}$)
• Vertical Shift: None

Step 6:

There is no step 6 provided in the original solution.

Knowledge Notes:

The problem involves analyzing the trigonometric function $y=5\tan (x-\frac{\pi }{3})$ to find its amplitude, period, and phase shift. Here are the relevant knowledge points:

1. Standard Form of Trigonometric Functions: The standard form for a tangent function is $y=a\tan (bx-c)+d$, where:

   • $a$ is the amplitude coefficient (though for tangent, amplitude is not applicable).

   • $b$ affects the period of the function.

   • $c$ determines the phase shift.

   • $d$ represents the vertical shift.

2. Amplitude: The amplitude of a function is the height from the center line to the peak or trough. For sine and cosine functions, this is a meaningful concept, but for tangent functions, which have no maximum or minimum, amplitude is not defined.

3. Period: The period of a trigonometric function is the length of one complete cycle. For tangent functions, the period is calculated as $\frac{\pi }{|b|}$, where $b$ is the coefficient from the standard form.

4. 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 that represents the horizontal shift in the standard form of the function.

5. Absolute Value: The absolute value of a number is its distance from zero on the number line, denoted as $|x|$. It is always non-negative.

6. Vertical Shift: The vertical shift moves the function up or down along the y-axis and is represented by the constant $d$ in the standard form. In this problem, $d=0$, so there is no vertical shift.

Understanding these concepts is crucial for analyzing and graphing trigonometric functions.