To determine the graph of the function [tex]\( y = 2 \tan \left( x + \frac{3\pi}{4} \right) \)[/tex], we need to perform a series of transformations on the basic graph of the tangent function, [tex]\( y = \tan(x) \)[/tex]. Here's a step-by-step explanation of how to transform the graph:
1. Horizontal Shift:
The expression inside the tangent function is [tex]\( x + \frac{3\pi}{4} \)[/tex]. This represents a horizontal shift. Specifically, [tex]\( \tan(x) \)[/tex] shifts to the left by [tex]\( \frac{3\pi}{4} \)[/tex].
2. Vertical Stretch:
The factor of 2 in front of the tangent function [tex]\( y = 2 \tan(x) \)[/tex] signifies a vertical stretch. Thus, every point on the graph of [tex]\( \tan(x) \)[/tex] is stretched vertically by a factor of 2. This means that the y-values are twice as far from the x-axis compared with the graph of [tex]\( y = \tan(x) \)[/tex].
### Step-by-Step Transformation:
1. Start with the basic graph of [tex]\( y = \tan(x) \)[/tex]:
- The graph has vertical asymptotes at [tex]\( x = \frac{\pi}{2} + k\pi \)[/tex], where [tex]\( k \)[/tex] is any integer.
- The function [tex]\( \tan(x) \)[/tex] has a period of [tex]\( \pi \)[/tex], repeating every [tex]\( \pi \)[/tex] units.
- The x-intercepts are at [tex]\( x = k\pi \)[/tex].
2. Apply the horizontal shift by [tex]\( \frac{3\pi}{4} \)[/tex] units to the left:
- The vertical asymptotes of the new function [tex]\( y = \tan \left( x + \frac{3\pi}{4} \right) \)[/tex] will now occur at [tex]\( x = -\frac{3\pi}{4} + \frac{\pi}{2} + k\pi = \frac{\pi}{4} + k\pi - \frac{3\pi}{4} \)[/tex].
- Simplifying, the vertical asymptotes will be at [tex]\( x = -\frac{\pi}{4} + k\pi \)[/tex].
- The x-intercepts of the function will be shifted to [tex]\( x = -\frac{3\pi}{4} + k\pi \)[/tex].
3. Apply the vertical stretch by a factor of 2:
- The points on the original graph of [tex]\( \tan(x) \)[/tex] that were at [tex]\( (x, y) \)[/tex] become [tex]\( (x, 2y) \)[/tex].
### Characteristics of the new function [tex]\( y = 2 \tan \left( x + \frac{3\pi}{4} \right) \)[/tex]:
- Period: The period remains [tex]\( \pi \)[/tex].
- Vertical Stretch Factor: The function now has a vertical stretch factor of 2.
- Vertical Asymptotes: They are at [tex]\( x = -\frac{\pi}{4} + k\pi \)[/tex], for any integer [tex]\( k \)[/tex].
- X-intercepts: They are at [tex]\( x = -\frac{3\pi}{4} + k\pi \)[/tex].
The graph has the same general shape as the tangent function, with vertical asymptotes, repeating every [tex]\( \pi \)[/tex] units, but it has been shifted and stretched. The graph will look like [tex]\( y = \tan(x) \)[/tex], but shifted left by [tex]\( \frac{3\pi}{4} \)[/tex] and stretched vertically by a factor of 2.
