Is there such a graph? A graph that increases at a decreasing rate with the graph approaching a slant asymptote as x decreases to negative infinity while the graph approaching a horizontal asymptote as x increases to positive infinity.
I know there is a way to estimate the graph equation by having this, e.g. $$y=\frac{ax^2+b}{cx+d}=\frac{a}{c}x+e+\frac{f}{cx+d}$$ where $$y=\frac{a}{c}x+e$$ is the slant asymptote(obtain through long division).
But to have horizontal asymptote, it is the same as the above except that the long division will obatin this instead $$y=\frac{ax+b}{cx+d}=\frac{a}{c}+\frac{f}{cx+d}$$ where long division will yield this $$y=\frac{a}{c}$$, which is a horizontal asymptote.
How do I combine the two above to obtain the desired graph of having both horizontal and slant asymptote?
You cannot have, with a single rational expression
$$y=\dfrac{P(x)}{Q(x)}=\dfrac{a_nx^n+\cdots}{b_px^p+\cdots} \ \ \ (1)$$
(with $n=p$ or $n=p+1$, the only cases of interest) a horizontal asymptote for $x\rightarrow +\infty$ and a slant asymptote for $x\rightarrow -\infty$.
This is due to the fact that, expression (1) has the same behaviour at $-\infty$ and $\infty$: it is equivalent to $\dfrac{a_n}{b_p}x^{n-p}$ : thus
if $n=p$, it is the horizontal asymptote with equation $y=\dfrac{a_n}{b_p}$ at both ends.
if $n=p+1$, it is the same slant asymptote at both ends (as you have explained in your question).
A different behavior at $-\infty$ and $-\infty$ needs the introduction of non-rational functions, for example
a square root like in $y=x-\sqrt{x^2+1}$ (which is a branch of hyperbola) or
an exponential like in $y=\dfrac{x}{1+e^{-x}}$ (see graphical representation below), etc...
(which, both, have a horizontal asymptote for $x\rightarrow +\infty$ and a slant asymptote for $x\rightarrow -\infty$, as you desire.