To graph the function [tex]\( y = 2 \tan \left( x + \frac{3\pi}{4} \right) \)[/tex], follow these steps:
### 1. Understand the Basic Tangent Function
First, recall the properties of the basic tangent function [tex]\( y = \tan(x) \)[/tex]:
- It has vertical asymptotes at [tex]\( x = \frac{\pi}{2} + k\pi \)[/tex] for any integer [tex]\( k \)[/tex].
- The function is periodic with period [tex]\( \pi \)[/tex].
- The tangent function has x-intercepts at [tex]\( x = k\pi \)[/tex].
### 2. Transformation of Tangent Function
Our function has two transformations applied to the basic tangent function:
1. Horizontal Shift: [tex]\( x + \frac{3\pi}{4} \)[/tex]
2. Vertical Scaling: Multiplying by 2.
### 3. Determine the Horizontal Shift
The term [tex]\( x + \frac{3\pi}{4} \)[/tex] indicates a horizontal shift. Specifically, this is a shift to the left by [tex]\( \frac{3\pi}{4} \)[/tex].
### 4. Establish Asymptotes
The vertical asymptotes of the function will be shifted according to the horizontal shift.
- Original vertical asymptotes for [tex]\( y = \tan(x) \)[/tex]: [tex]\( x = \frac{\pi}{2} + k\pi \)[/tex].
- Applying the shift [tex]\( x \rightarrow x + \frac{3\pi}{4} \)[/tex]: [tex]\( x + \frac{3\pi}{4} = \frac{\pi}{2} + k\pi \)[/tex].
Solving for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{\pi}{2} - \frac{3\pi}{4} + k\pi \][/tex]
[tex]\[ x = -\frac{\pi}{4} + k\pi \][/tex]
So the vertical asymptotes for our function are at [tex]\( x = -\frac{\pi}{4} + k\pi \)[/tex].
### 5. Determine the Period
The period of the tangent function is not changed by the horizontal shift, so the period remains [tex]\( \pi \)[/tex].
### 6. Vertical Scaling
The factor of 2 vertically scales the tangent function, stretching it by a factor of 2.
### 7. Sketching the Graph
1. Asymptotes: Draw vertical asymptotes at [tex]\( x = -\frac{\pi}{4} + k\pi \)[/tex]. For example, at [tex]\( -\frac{\pi}{4} \)[/tex], [tex]\( \frac{3\pi}{4} \)[/tex], [tex]\( \frac{7\pi}{4} \)[/tex], etc.
2. Tangent Behavior: Between each pair of consecutive asymptotes, draw the basic shape of the tangent function, but stretched vertically.
3. X-Intercepts: The x-intercepts will occur at the points halfway between asymptotes, which are [tex]\( x = -\frac{\pi}{4} + \frac{\pi}{2} + k\pi = \frac{\pi}{4} + k\pi \)[/tex].
Thus, the graph of [tex]\( y = 2 \tan \left( x + \frac{3\pi}{4} \right) \)[/tex] will consist of the basic tangent shape, stretched vertically by a factor of 2, with vertical asymptotes at [tex]\( x = -\frac{\pi}{4} + k\pi \)[/tex] and x-intercepts at [tex]\( x = \frac{\pi}{4} + k\pi \)[/tex].
