Question
Identify the vertical stretching factor and period of the function. Answer exactly. \( f(x)=-2 \cot (5 x) \) Vertical stretching factor: Period:
Ask by Wyatt Washington. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
Vertical stretching factor: 2
Period: \(\dfrac{\pi}{5}\)
Solution
For the function \( f(x) = -2 \cot(5x) \):
1. **Vertical Stretching Factor:**
The vertical stretching factor is determined by the absolute value of the coefficient multiplying the cotangent function. Here, the coefficient is \(-2\), so the vertical stretching factor is:
\[
\text{Vertical stretching factor} = 2
\]
2. **Period:**
The period of the cotangent function \( \cot(x) \) is \( \pi \). When the function is of the form \( \cot(Bx) \), the period is adjusted by the factor \( B \). Therefore, for \( \cot(5x) \), the period \( T \) is:
\[
T = \frac{\pi}{5}
\]
**Final Answer:**
- **Vertical stretching factor:** \(2\)
- **Period:** \(\dfrac{\pi}{5}\)
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Simplify this solution Mind Expander
The vertical stretching factor of the function \( f(x)=-2 \cot (5 x) \) is the absolute value of the coefficient in front of the cotangent function, which is 2. The period of the cotangent function can be found using the formula \( \frac{\pi}{|b|} \), where \( b \) is the coefficient of \( x \). In this case, \( b = 5 \), so the period is \( \frac{\pi}{5} \).
