How to find period from equation?
Understand the Problem
The question is asking how to determine the period of a function given its equation. To solve this, we will identify the type of function (like sinusoidal or periodic functions) and use the appropriate mathematical formula to find the period.
Answer
The period of the function is $P = \frac{2\pi}{|k|}$.
Answer for screen readers
The period of the given function is $P = \frac{2\pi}{|k|}$.
Steps to Solve
- Identify the Type of Function
Check the provided function equation to determine if it is a sinusoidal function, such as a sine or cosine function. For example, $f(x) = \sin(kx)$ or $f(x) = \cos(kx)$.
- Determine the Coefficient of x
Identify the coefficient $k$ in the function's equation. For instance, if the function is $f(x) = \sin(3x)$, then $k = 3$.
- Calculate the Period Using the Formula
The period $P$ of a sinusoidal function can be calculated using the formula:
$$ P = \frac{2\pi}{|k|} $$
where $|k|$ is the absolute value of the coefficient of $x$.
- Substitute the Coefficient into the Formula
Substitute the value of $k$ into the period formula. Using the previous example where $k = 3$, we have:
$$ P = \frac{2\pi}{3} $$
- Simplify the Period Answer
If possible, simplify the period for clarity. In the example, $P$ is already in its simplest form as $\frac{2\pi}{3}$.
The period of the given function is $P = \frac{2\pi}{|k|}$.
More Information
The period of sinusoidal functions is a fundamental concept in trigonometry. It represents the length of one complete cycle of the wave. For example, the sine and cosine functions have a default period of $2\pi$. If the coefficient $k$ is changed, it affects how quickly the function oscillates.
Tips
- Confusing period with frequency: Remember that the period is the time it takes to complete one cycle, while frequency is the number of cycles per unit time.
- Not taking the absolute value of $k$: Always use the absolute value of the coefficient of $x$ in the formula for the period.
