The given function is [tex]\( y = 2 \tan \left( x + \frac{3\pi}{4} \right) \)[/tex]. Let's break down and understand how this function transforms the standard tangent function step by step.
1. Understanding the Basic Tangent Function:
- The basic tangent function is [tex]\( y = \tan(x) \)[/tex].
- Its graph has vertical asymptotes at [tex]\( x = \frac{\pi}{2} + k\pi \)[/tex], where [tex]\( k \)[/tex] is an integer.
- It has a period of [tex]\( \pi \)[/tex], meaning the function repeats itself every [tex]\( \pi \)[/tex] units along the x-axis.
- The basic shape of the tangent function includes increasing from [tex]\(-\infty\)[/tex] to [tex]\(\infty\)[/tex] between the asymptotes.
2. Horizontal Shift:
- The given function includes a term [tex]\( x + \frac{3\pi}{4} \)[/tex] inside the tangent function.
- This represents a horizontal shift. Specifically, the graph will shift to the left by [tex]\( \frac{3\pi}{4} \)[/tex].
- The new vertical asymptotes, previously at [tex]\( x = \frac{\pi}{2} + k\pi \)[/tex], will shift to [tex]\( x = \frac{\pi}{2} - \frac{3\pi}{4} + k\pi \)[/tex].
Simplifying this, we get the new asymptotes at [tex]\( x = -\frac{\pi}{4} + k\pi \)[/tex].
3. Vertical Stretch:
- The coefficient 2 in front of the tangent function scales the graph vertically.
- This means the output values (y-values) of the tangent function are stretched by a factor of 2.
- The basic shape of the graph remains the same, but it will be twice as tall at any given x-value.
4. Comprehensive Description:
- The final graph of [tex]\( y = 2 \tan \left( x + \frac{3\pi}{4} \right) \)[/tex] will still have a period of [tex]\( \pi \)[/tex].
- Due to the horizontal shift, the vertical asymptotes will now be located at [tex]\( x = -\frac{\pi}{4} + k\pi \)[/tex].
- The graph will intersect the x-axis at [tex]\( x = -\frac{3\pi}{4} + k\pi \)[/tex], which are the points where the original [tex]\(\tan\left(x + \frac{3\pi}{4}\right)\)[/tex] function equals zero.
- The output values for each value of [tex]\( x \)[/tex] will be scaled by 2, making the graph taller.
### Summary:
The graph of [tex]\( y = 2 \tan \left( x + \frac{3\pi}{4} \right) \)[/tex] is a horizontally shifted and vertically scaled version of the standard tangent function. It will have the same general shape as the basic tangent function but will be shifted to the left by [tex]\( \frac{3\pi}{4} \)[/tex] units and vertically stretched by a factor of 2. The vertical asymptotes will be at [tex]\( x = -\frac{\pi}{4} + k\pi \)[/tex], and the graph will cross the x-axis at [tex]\( x = -\frac{3\pi}{4} + k\pi \)[/tex].
