How do you write an equation of the tangent function with period $ \dfrac{\pi }{4} $ , phase shift $ \pi $ , and vertical shift 1?

Hint: In this question, we have to find an equation that satisfies the given statement. We have to write an equation of the tangent function, that is we have to convert a statement into a mathematical equation. As we know, a tangent function is a periodic function, and a periodic function is a function that repeats its value at a regular interval. The standard form of the tangent equation is $ y=A\tan (Bx-C)+D $ where $ period=\dfrac{\pi }{|B|}, $ $ phase\text{ }shift=\dfrac{-C}{B}, $ and $ vertical\text{ }shift=D $ . So, we first find the value of the constants of the equation with the help of period, phase shift, and vertical shift. And then, put the value of constants in the equation, to get the required result of the solution.

Complete step by step answer:
According to the question, we have to find the equation of a tangent function.
The statement given to us is the tangent function with period $ \dfrac{\pi }{4} $ , phase shift $ \pi $ , and vertical shift 1 .
So, we will use the period, phase shift, and vertical shift to find the value of constants.
The standard form of the tangent equation is $ y=A\tan (Bx-C)+D $ --------- (1)
Now, we know that $ period=\dfrac{\pi }{|B|} $ and the value of a period given in the statement is $ \dfrac{\pi }{4} $ , therefore we equate both the values, that is
 $ \dfrac{\pi }{4}=\dfrac{\pi }{|B|} $
Now, multiply |B| on both sides in the above equation, we get
 $ \dfrac{\pi }{4}.|B|=\dfrac{\pi }{|B|}.|B| $
On further solving, we get
 $ \dfrac{\pi }{4}.|B|=\pi $
Now, we will again multiply $ \dfrac{4}{\pi } $ on both sides in the above equation, we get
 $ \dfrac{\pi }{4}.|B|.\dfrac{4}{\pi }=\pi .\dfrac{4}{\pi } $
Therefore, we get
 $ |B|=4 $ --------- (2)
Now, we will find the value of C using phase-shift.
As we know, the $ phase\text{ }shift=\dfrac{-C}{B}, $ and the value of phase shift given in the question is $ \pi $ , therefore on equating both the values, we get
 $ -\dfrac{C}{B}=\pi $
Now, we will multiply B on both sides in the above equation, we get
 $ -\dfrac{C}{B}.B=\pi .B $
On further simplification, we get
 $ -C=B\pi $
Again, we will multiply (-1) on both sides in the above equation, we get
 $ -C.(-1)=B\pi .(-1) $
Therefore, we get
 $ C=-B\pi $
Now, we will put the value of equation (2) in the above equation, we get
 $ C=-4\pi $ --------- (3)
As we know, the $ vertical\text{ }shift=D $ , and the given value of vertical shift in the statement is 1, therefore on comparing both the values, we get
 $ D=1 $ ------- (4)
So, we know put the value of equations (2), (3), and (4) in equation (1), we get
 $ y=\tan (4x-4\pi )+1 $
On further simplification, we get
$\Rightarrow$ $ y=\tan 4(x-\pi )+1 $
$\Rightarrow$ Therefore, for the statement the tangent function with period $ \dfrac{\pi }{4} $ , phase shift $ \pi $ , and vertical shift 1, its mathematical equation is $ y=\tan 4(x-\pi )+1 $ .

Note:
 While solving this problem, do mention all the formulas in the steps to avoid confusion and mathematical errors. Also, do remember that the value of constant A is equal to the amplitude. Since we do not have any value of amplitude in the question, so we simply put it equals 1, that is $ A=1 $.