At Westonci.ca, we connect you with experts who provide detailed answers to your most pressing questions. Start exploring now! Discover reliable solutions to your questions from a wide network of experts on our comprehensive Q&A platform. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our 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].
Thank you for trusting us with your questions. We're here to help you find accurate answers quickly and efficiently. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.
