Discover the answers you need at Westonci.ca, where experts provide clear and concise information on various topics. Discover reliable solutions to your questions from a wide network of experts on our comprehensive Q&A platform. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
Sagot :
To graph the function [tex]\( y = 0.25 \csc (x + \pi) + 1 \)[/tex], we'll need to consider several factors. Let's break it down step-by-step:
1. Understanding [tex]\( \csc \)[/tex] Operator: The cosecant function, [tex]\( \csc(x) \)[/tex], is the reciprocal of the sine function:
[tex]\[ \csc(x) = \frac{1}{\sin(x)} \][/tex]
So, [tex]\( y = 0.25 \csc (x + \pi) + 1 \)[/tex] will have vertical asymptotes where [tex]\(\sin(x + \pi) = 0\)[/tex].
2. Periodicity of [tex]\( \csc(x + \pi) \)[/tex]: The sine function [tex]\( \sin(x + \pi) \)[/tex] has the same period as [tex]\( \sin(x) \)[/tex], which is [tex]\(2\pi \)[/tex]. Therefore, [tex]\( \csc(x + \pi) \)[/tex] will also repeat every [tex]\(2\pi\)[/tex].
3. Horizontal Shift: The term [tex]\( x + \pi \)[/tex] indicates a phase shift to the left by [tex]\(\pi\)[/tex]. This means that the graph of [tex]\( \csc(x) \)[/tex] is shifted to the left by [tex]\(\pi\)[/tex].
4. Amplitude and Vertices: The coefficient [tex]\( 0.25 \)[/tex] affects the amplitude of [tex]\( \csc(x + \pi) \)[/tex]. The [tex]\(0.25 \csc(x + \pi)\)[/tex] means all the values of [tex]\( \csc(x + \pi) \)[/tex] will be multiplied by [tex]\( 0.25 \)[/tex]. Therefore, the vertical stretch of the basic cosecant function is reduced by a factor of 4.
5. Vertical Shift: The addition of 1 means that the entire graph of [tex]\(0.25 \csc(x + \pi)\)[/tex] is shifted upward by 1 unit.
6. Plot the Function:
- Vertical Asymptotes: Determine where [tex]\( \sin(x + \pi) = 0 \)[/tex]. This occurs at [tex]\( x = -\pi, 0, \pi, 2\pi, \ldots\)[/tex].
- Key Points: Calculate [tex]\( \csc \)[/tex]'s behavior between these points.
- Transformations: Apply the amplitude, phase shift, and vertical shift.
- At points where [tex]\( \sin(x + \pi) = 1 \)[/tex], the values of [tex]\( 0.25 \csc(x + \pi) \)[/tex] will be [tex]\( \pm 0.25 \)[/tex]. Adding 1 gives points at [tex]\(y = 1.25\)[/tex].
So, a step-by-step approach will help identify the general shape and features of the graph:
- Plot vertical asymptotes at [tex]\( x = -\pi, 0, \pi, 2\pi, \ldots \)[/tex];
- Between asymptotes, plot the general [tex]\( \csc \)[/tex] shape, compressing by factor 0.25 and shifting up by 1 unit.
By following these steps, you can sketch the function [tex]\( y = 0.25 \csc (x + \pi) + 1 \)[/tex].
1. Understanding [tex]\( \csc \)[/tex] Operator: The cosecant function, [tex]\( \csc(x) \)[/tex], is the reciprocal of the sine function:
[tex]\[ \csc(x) = \frac{1}{\sin(x)} \][/tex]
So, [tex]\( y = 0.25 \csc (x + \pi) + 1 \)[/tex] will have vertical asymptotes where [tex]\(\sin(x + \pi) = 0\)[/tex].
2. Periodicity of [tex]\( \csc(x + \pi) \)[/tex]: The sine function [tex]\( \sin(x + \pi) \)[/tex] has the same period as [tex]\( \sin(x) \)[/tex], which is [tex]\(2\pi \)[/tex]. Therefore, [tex]\( \csc(x + \pi) \)[/tex] will also repeat every [tex]\(2\pi\)[/tex].
3. Horizontal Shift: The term [tex]\( x + \pi \)[/tex] indicates a phase shift to the left by [tex]\(\pi\)[/tex]. This means that the graph of [tex]\( \csc(x) \)[/tex] is shifted to the left by [tex]\(\pi\)[/tex].
4. Amplitude and Vertices: The coefficient [tex]\( 0.25 \)[/tex] affects the amplitude of [tex]\( \csc(x + \pi) \)[/tex]. The [tex]\(0.25 \csc(x + \pi)\)[/tex] means all the values of [tex]\( \csc(x + \pi) \)[/tex] will be multiplied by [tex]\( 0.25 \)[/tex]. Therefore, the vertical stretch of the basic cosecant function is reduced by a factor of 4.
5. Vertical Shift: The addition of 1 means that the entire graph of [tex]\(0.25 \csc(x + \pi)\)[/tex] is shifted upward by 1 unit.
6. Plot the Function:
- Vertical Asymptotes: Determine where [tex]\( \sin(x + \pi) = 0 \)[/tex]. This occurs at [tex]\( x = -\pi, 0, \pi, 2\pi, \ldots\)[/tex].
- Key Points: Calculate [tex]\( \csc \)[/tex]'s behavior between these points.
- Transformations: Apply the amplitude, phase shift, and vertical shift.
- At points where [tex]\( \sin(x + \pi) = 1 \)[/tex], the values of [tex]\( 0.25 \csc(x + \pi) \)[/tex] will be [tex]\( \pm 0.25 \)[/tex]. Adding 1 gives points at [tex]\(y = 1.25\)[/tex].
So, a step-by-step approach will help identify the general shape and features of the graph:
- Plot vertical asymptotes at [tex]\( x = -\pi, 0, \pi, 2\pi, \ldots \)[/tex];
- Between asymptotes, plot the general [tex]\( \csc \)[/tex] shape, compressing by factor 0.25 and shifting up by 1 unit.
By following these steps, you can sketch the function [tex]\( y = 0.25 \csc (x + \pi) + 1 \)[/tex].
Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.
