  We study a one-parameter family of interval maps
$\{T_\alpha\}_{\alpha\in[1,\beta]}$, with $\beta$ the golden mean, defined on
$[-1,1]$ by $T_\alpha(x)=\beta^{1+|t|}x-t\beta\alpha$ where $t\in\{-1,0,1\}$.
For each $T_\alpha,\ \alpha>1$, we construct its unique, absolutely continuous
invariant measure and show that on an open, dense subset of parameters
$\alpha$, the corresponding density is a step function with finitely many
jumps. We give an explicit description of the maximal intervals of parameters
on which the density has at most the same number of jumps. A main tool in our
analysis is the phenomenon of matching, where the orbits of the left and right
limits of discontinuity points meet after a finite number of steps. Each
$T_\alpha$ generates signed expansions of numbers in base $1/\beta$; via
Birkhoff's ergodic theorem, the invariant measures are used to determine the
asymptotic relative frequencies of digits in generic $T_\alpha$-expansions. In
particular, the frequency of $0$ is shown to vary continuously as a function of
$\alpha$ and to attain its maximum $3/4$ on the maximal interval
$[1/2+1/\beta,1+1/\beta^2]$.
