MA(1) ACVF and ACF
Model
$$X_t = Z_t + \theta Z_{t-1}, \qquad Z_t \sim \text{WN}(0, \sigma^2)$$ACVF Derivation
$E(X_t) = 0$.
$\gamma(0) = \text{Var}(X_t) = \text{Var}(Z_t) + \theta^2 \text{Var}(Z_{t-1}) = (1 + \theta^2)\sigma^2$
$\gamma(1) = \text{Cov}(X_{t+1}, X_t) = \text{Cov}(Z_{t+1} + \theta Z_t,\; Z_t + \theta Z_{t-1})$
The only overlapping noise term is $Z_t$ (appears in both with coefficients $\theta$ and $1$):
$$\gamma(1) = \theta \cdot \text{Var}(Z_t) = \theta \sigma^2$$$\gamma(2) = \text{Cov}(Z_{t+2} + \theta Z_{t+1},\; Z_t + \theta Z_{t-1}) = 0$ (no overlapping $Z$ subscripts)
$$\gamma(h) = \begin{cases} (1+\theta^2)\sigma^2, & h = 0 \\ \theta\sigma^2, & |h| = 1 \\ 0, & |h| \geq 2 \end{cases}$$ACF
$$\rho(h) = \begin{cases} 1, & h = 0 \\ \frac{\theta}{1+\theta^2}, & |h| = 1 \\ 0, & |h| \geq 2 \end{cases}$$Summary
MA(1) is stationary. ACF cuts off after lag 1.
Computation Technique
When computing $\gamma(h) = \text{Cov}(X_{t+h}, X_t)$, expand both $X_{t+h}$ and $X_t$ in terms of $Z$, then look for overlapping subscripts — only those contribute (since $\text{Cov}(Z_i, Z_j) = 0$ for $i \neq j$).