For the fitted line $\hat y_0=\hat\beta_0+\hat\beta_1x_0$ the variance of the estimated mean response at $x_0$ is

$\operatorname{Var}(\hat y_0)=\sigma^{2}\left[\frac{1}{n} +\frac{(x_0-\overline{x})^{2}}{S_{xx}}\right], \qquad S_{xx}=\sum_i(x_i-\overline{x})^{2}.$

The expression depends on $x_0$ only through $(x_0-\overline{x})^{2}$ , and increases with it. So precision is greatest at the centre of the data ( $x_0=\overline{x}$ , where the variance is the minimum $\sigma^{2}/n$ ) and degrades symmetrically as you move away in either direction — which is why confidence bands about a regression line are hyperbolic, flaring out at both ends.

That rules out (B), which denies any dependence on $x_0$ , and (C), which gets the geometry wrong: a value far below $\overline{x}$ is predicted just as imprecisely as one equally far above. (A) is simply false — the prediction is a function of $x_0$ .

The practical moral is the danger of extrapolation: predictions far from $\overline{x}$ carry rapidly inflating variance.