To determine the graph of the function \( y = 0.25 \csc(x + \pi) + 1 \), we need to understand the properties and behavior of the \( \csc \) (cosecant) function and how the transformations affect its graph.
The cosecant function, \( \csc(x) \), is the reciprocal of the sine function, \( \csc(x) = \frac{1}{\sin(x)} \). This function has its own characteristics:
1. Asymptotes: The \( \csc(x) \) function has vertical asymptotes where \( \sin(x) = 0 \), which occurs at \( x = k\pi \) for any integer \( k \).
2. Period: The period of the \( \csc(x) \) function is \( 2\pi \), meaning the function repeats its pattern every \( 2\pi \) units.
3. Range: \( \csc(x) \) is defined wherever \( \sin(x) \) is non-zero and takes values \( (-\infty, -1] \cup [1, \infty) \).
With this knowledge, we can transform the \( \csc(x) \) function to match the given equation:
### Step-by-Step Transformation:
1. Phase Shift: The expression \( \csc(x + \pi) \) shifts the graph of \( \csc(x) \) to the left by \( \pi \) units. This means the vertical asymptotes, which originally are at \( x = k\pi \), will now be at \( x = k\pi - \pi \) or \( x = (k-1)\pi \).
2. Vertical Scaling: The function \( 0.25 \csc(x + \pi) \) compresses the graph vertically. The typical values of \( \csc(x) \) that are normally \( \pm1, \pm2, \ldots \) become \( \pm0.25, \pm0.5, \ldots \)
3. Vertical Translation: Adding 1 to \( 0.25 \csc(x + \pi) \) shifts the curve up by 1 unit. Thus, the whole graph moves upward by 1 unit.
### Constructing the Graph:
- Vertical Asymptotes: After the phase shift, vertical asymptotes will occur where \( x + \pi \) is a multiple of \( \pi \). This translates to \( x = (k-1)\pi \) for any integer \( k \).
- Range: Before adding 1, the range of \( 0.25 \csc(x + \pi) \) is \( (-\infty, -0.25] \cup [0.25, \infty) \). Adding 1 shifts this to \( (-\infty, 0.75] \cup [1.25, \infty) \).
### Key Points:
- The \( x \)-values where the function approaches vertical asymptotes are \( (k-1)\pi \) for any integer \( k \).
- Between these asymptotes, the graph will have a 'hyperbolic' shape, reaching its minimum or maximum halfway between the asymptotes.
- The curve's values oscillate around the vertical stretch and shift, meaning it peaks away from the horizontal line shifted up by 1 unit.
### Summary of the Graph:
- It has vertical asymptotes at \( x = (k-1)\pi \).
- It oscillates such that its minimum and maximum points are shifted up by 1 unit.
- Between the asymptotes, the function has the values compressed by a factor of 0.25 and shifted up by 1.
The graph of \( y = 0.25 \csc(x + \pi) + 1 \) will look like a series of upward and downward curving branches centered around vertical asymptotes but all shifted up 1 unit and compressed by a factor of 0.25.
It would be a good exercise to sketch it or use graphing technology to visualize this transformed function.
