I am unfortunately lost on this question and would really like some help!
Question:
The graph of $f(x)=\frac{a(x-b)(x-c)^2}{(x-d)(x-k)^2}$ , where a, b, c, d, and k are integer constants, is shown in three different graphs. Determine the value of all five constants based on the characteristics of the graph. Explicitly state the value of each constant.
One of the graphs provided for solving:
$x=-1$ is a multiple root since it's also a local maximum. (It must be an even order root since the function doesn't change sign.) This means $c=-1$.
The other root $x=5$ is a simple root (or, in general, of odd order) since the function changes sign. Therefore $b=5$.
The lateral limits at $x=3$ have the same sign, so that must be a double (or, in general, even order) zero of the denominator, thus $k=3$.
The lateral limits at $x=-2$ have opposite signs, so that's a simple (or, in general, odd order) zero of the denominator, so $d=-2$.
Thus far it's been determined that $f(x)=a \frac{(x-5)(x+1)^2}{(x+2)(x-3)^2}$. To find the constant $a$ note that $f(0) = \frac{-5a}{18}$ is close to but slightly larger than $-1$. Since $a$ is an integer this gives $a=3$.
For verification, you can check the complete graph in Wolfram Alpha and confirm that $f(x)$ has a local maximum at around $x=14$ (which is outside your graph) then it decreases towards the horizontal asymptote $y=3$ for $x \to \infty$.