Find Amplitude, Period, and Phase Shift y=5tan(x-pi/3)
The problem is asking for the identification of three specific characteristics of the trigonometric function provided, which is a tangent function. The characteristics to be determined are:
Amplitude: In the context of tangent functions, amplitude typically isn't defined in the same way as it is for sine or cosine functions, because the tangent function does not have a maximum or minimum value. However, in a broader sense, you might consider any factor affecting the "steepness" or "scale" of the graph. The question suggests analyzing the coefficient of the tangent function to describe this characteristic.
Period: The period of a trigonometric function is the length of the interval over which it repeats. For the standard tangent function, this is π. The problem is asking to determine if and how the period has been altered by the function given.
Phase Shift: The phase shift of a trigonometric function is the horizontal displacement of its graph from the origin or from its usual position. This shift is usually a result of a constant being added or subtracted within the function's argument—in this case, (x - π/3). The task is to calculate how this affects the starting point or phase of the basic tangent function.
The expression provided is y = 5tan(x - π/3), and it is the modification of the basic tan(x) function with potential alterations to its amplitude (again, interpreted in terms of steepness for the tangent function), period, and phase shift, which need to be determined based on the given transformation.
$y = 5 tan \left(\right. x - \frac{\pi}{3} \left.\right)$
Answer
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:
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.
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.
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.
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.
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.
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.
