How do I determine the value of $d$ from the tangent graph of the form $y = a\times \tan (bx + c) + d$? 
I understand that $d$ is the vertical shift, and I have already worked out the values of $a$, $b$ and $c$ to be the following (please tell me if I am wrong):
$\pmb{a}\quad$ Given point is $(-1, -7)$ so I substituted it in as the values of  $x$ and $y$ ; $-7= a\times \tan\left(\dfrac{\pi}{4}\left(-1\right)+5\right)$ which I solved for $a$ and that gave me$\space$ -3.8040.
$\pmb{b}\quad$ Period so $\dfrac{\pi}{4}$
$\pmb{c}\quad$ $y$-intercept so $5$
But how do I find $d$ for this type of graph with only vertical asymptotes?

You are correct in that $b = \frac{\pi}{4}$, but your reasoning isn't so clear. The function's period is $4$, and since $f(x) = \tan (\alpha x)$ has period $\pi / \alpha$, $b$ is $\frac{\pi}{4}$ by simple arithmetic.
So far we have $f(x) = a\tan(\frac{\pi}{4}x + c) + d$.
The function $g(x) = \tan(\frac{\pi}{4}x)$ has asymptotes at $x = 4n + 2$ e.g $x = 2$ where $n$ is an integer, while $f(x)$ has asymptotes at $x = 4n +1$. This means in the simplest case that $f(x)$ is $\tan(\frac{\pi}{4}x)$ translated $1$ unit to the left.
Hence $$ f(x) = a\tan\left(\frac{\pi}{4}(x+1)\right) + d \tag{1}\label{eq1}$$
Substitute the value $f(-1) = -7$ into \eqref{eq1} to obtain that $d = -7$, and substitute $f(0) = -5$ to obtain $a = 2$.
Finally we arrive at
$$f(x) = 2 \tan \left(\frac{\pi}{4}x + \frac{\pi}{4}\right) -7$$
I hope this answer was clear.
This is my first time giving an answer on this website, if you are a more experienced user and have any constructive criticisms feel free to share them with me.
