  Let $K$ be a number field, and let $C$ be a hyperelliptic curve over $K$ with
Jacobian $J$. Suppose that $C$ is defined by an equation of the form $y^{2} =
f(x)(x - \lambda)$ for some irreducible monic polynomial $f \in
\mathcal{O}_{K}[x]$ of discriminant $\Delta$ and some element $\lambda \in
\mathcal{O}_{K}$. Our first main result says that if there is a prime
$\mathfrak{p}$ of $K$ dividing $(f(\lambda))$ but not $(2\Delta)$, then the
image of the natural $2$-adic Galois representation is open in
$\mathrm{GSp}(T_{2}(J))$ and contains a certain congruence subgroup of
$\mathrm{Sp}(T_{2}(J))$ depending on the maximal power of $\mathfrak{p}$
dividing $(f(\lambda))$. We also present and prove a variant of this result
that applies when $C$ is defined by an equation of the form $y^{2} = f(x)(x -
\lambda)(x - \lambda')$ for distinct elements $\lambda, \lambda' \in K$. We
then show that the hypothesis in the former statement holds for almost all
$\lambda \in \mathcal{O}_{K}$ and prove a quantitative form of a uniform
boundedness result of Cadoret and Tamagawa.
