To sketch two periods of the graph of the function [tex]\( p(x) = \tan\left(x - \frac{\pi}{4}\right) \)[/tex], we'll need to identify its stretching factor, period, and asymptotes.
### Stretching Factor:
The function [tex]\( \tan\left(x - \frac{\pi}{4}\right) \)[/tex] does not have a coefficient that modifies the tangent term directly (in the form of [tex]\( A \cdot \tan(B(x - C)) \)[/tex]). Hence, the stretching factor of the tangent function remains:
[tex]\[
\text{Stretching factor} = 1
\][/tex]
### Period:
The period of the tangent function [tex]\( \tan(x) \)[/tex] is [tex]\( \pi \)[/tex]. For the function [tex]\( p(x) = \tan\left(x - \frac{\pi}{4}\right) \)[/tex], the horizontal shift (in this case, [tex]\(\frac{\pi}{4}\)[/tex]) does not affect the period. Therefore, the period of [tex]\( p(x) \)[/tex] is:
[tex]\[
\text{Period} = \pi
\][/tex]
### Asymptotes:
The vertical asymptotes of the standard tangent function occur where the argument of the tangent function is of the form [tex]\( \frac{\pi}{2} + k\pi \)[/tex], where [tex]\( k \)[/tex] is an integer. For the function [tex]\( p(x) = \tan\left(x - \frac{\pi}{4}\right) \)[/tex], the argument will be undefined where:
[tex]\[
x - \frac{\pi}{4} = \frac{\pi}{2} + k\pi
\][/tex]
Solving for [tex]\( x \)[/tex], we get:
[tex]\[
x = \frac{\pi}{4} + \frac{\pi}{2} + k\pi = \frac{\pi}{4} + \left(2k+1\right)\frac{\pi}{2}
\][/tex]
By considering [tex]\( k = -1, 0, 1 \)[/tex] (covering two periods around the origin), the asymptotes within the domain [tex]\([-P, P]\)[/tex] where [tex]\( P = \pi \)[/tex] are:
[tex]\[
x = -\frac{\pi}{4}, \quad x = \frac{3\pi}{4}, \quad x = \frac{7\pi}{4}
\][/tex]
These intervals are correctly placed within two periods because:
[tex]\[
- \frac{\pi}{4} \][/tex]
is less than [tex]\[((2 - 1)·pi / 2\) = ((1·Pi)/2)
\[= \pi\][/tex]
\[
\frac{3\pi}{4}, \space approximately
x = \frac{\pi}{4} + 1*pi/2. roughly and right at
\frac{7\pi}{4}, \ (but not exactly at).
Thus we get the following approximation values noted below as
close intervals within approximately these domains once we correct small fractions into a rounded set.
So, the exact coordinates will be noted as:
### Answers
- Stretching factor [tex]\( = 1 \)[/tex]
- Period: [tex]\( P = \pi \)[/tex]
- Asymptotes: [tex]\( x=-\frac{\pi}{4}; \ x = \frac{3\pi}{4}; x = \frac{7\pi}{4} \)[/tex].
