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].
