Westonci.ca is your trusted source for finding answers to all your questions. Ask, explore, and learn with our expert community. Get quick and reliable solutions to your questions from a community of experienced experts on our platform. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
Sagot :
To determine the period of the function [tex]\( y = \tan \left[\frac{1}{4}\left(x - \frac{\pi}{2}\right)\right] + 1 \)[/tex], we need to analyze the transformation that has been applied to the standard tangent function.
1. Standard Tangent Function:
- The period of the standard tangent function [tex]\( y = \tan(x) \)[/tex] is [tex]\( \pi \)[/tex].
2. Transformation Analysis:
- The given function is [tex]\( y = \tan \left[\frac{1}{4}\left(x - \frac{\pi}{2}\right)\right] + 1 \)[/tex].
- The argument of the tangent function is [tex]\( \frac{1}{4}\left(x - \frac{\pi}{2}\right) \)[/tex].
3. Coefficient Adjustment:
- For a general tangent function of the form [tex]\( y = \tan(bx + c) \)[/tex], the period is given by [tex]\( \frac{\pi}{|b|} \)[/tex].
- Here, the function’s argument is [tex]\( \frac{1}{4}(x - \frac{\pi}{2}) \)[/tex], which can be expressed as [tex]\( \frac{1}{4}x - \frac{\pi}{8} \)[/tex].
- The coefficient [tex]\( b \)[/tex] of [tex]\( x \)[/tex] in [tex]\( \frac{1}{4}x - \frac{\pi}{8} \)[/tex] is [tex]\( \frac{1}{4} \)[/tex].
4. Period Calculation:
- To find the period, we divide [tex]\( \pi \)[/tex] by the absolute value of the coefficient of [tex]\( x \)[/tex]:
[tex]\[ \text{Period} = \frac{\pi}{\left|\frac{1}{4}\right|} = \frac{\pi}{\frac{1}{4}} = 4\pi \][/tex]
Therefore, the period of the function [tex]\( y = \tan \left[\frac{1}{4}(x - \frac{\pi}{2})\right] + 1 \)[/tex] is [tex]\(\boxed{4\pi}\)[/tex].
1. Standard Tangent Function:
- The period of the standard tangent function [tex]\( y = \tan(x) \)[/tex] is [tex]\( \pi \)[/tex].
2. Transformation Analysis:
- The given function is [tex]\( y = \tan \left[\frac{1}{4}\left(x - \frac{\pi}{2}\right)\right] + 1 \)[/tex].
- The argument of the tangent function is [tex]\( \frac{1}{4}\left(x - \frac{\pi}{2}\right) \)[/tex].
3. Coefficient Adjustment:
- For a general tangent function of the form [tex]\( y = \tan(bx + c) \)[/tex], the period is given by [tex]\( \frac{\pi}{|b|} \)[/tex].
- Here, the function’s argument is [tex]\( \frac{1}{4}(x - \frac{\pi}{2}) \)[/tex], which can be expressed as [tex]\( \frac{1}{4}x - \frac{\pi}{8} \)[/tex].
- The coefficient [tex]\( b \)[/tex] of [tex]\( x \)[/tex] in [tex]\( \frac{1}{4}x - \frac{\pi}{8} \)[/tex] is [tex]\( \frac{1}{4} \)[/tex].
4. Period Calculation:
- To find the period, we divide [tex]\( \pi \)[/tex] by the absolute value of the coefficient of [tex]\( x \)[/tex]:
[tex]\[ \text{Period} = \frac{\pi}{\left|\frac{1}{4}\right|} = \frac{\pi}{\frac{1}{4}} = 4\pi \][/tex]
Therefore, the period of the function [tex]\( y = \tan \left[\frac{1}{4}(x - \frac{\pi}{2})\right] + 1 \)[/tex] is [tex]\(\boxed{4\pi}\)[/tex].
We hope our answers were useful. Return anytime for more information and answers to any other questions you have. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.