Although the graph is not explicitly provided, the options imply a plot that yields a straight line parallel to the $x$-axis (i.e., $y = \text{constant}$).

1. For a zero-order reaction, the rate equation is $\text{Rate} = k[R]^0 = k$. This means the rate is constant and independent of the concentration of the reactant. Thus, a plot of rate ($y$-axis) versus concentration ($x$-axis) is a straight horizontal line.
2. For a first-order reaction, the half-life is given by $t_{1/2} = \frac{0.693}{k}$ . This means the half-life is a constant value and independent of the initial concentration of the reactant . Thus, a plot of $t_{1/2}$ ($y$-axis) versus concentration ($x$-axis) is also a straight horizontal line. Therefore, the correct axes corresponding to a constant $y$-value are: zero order ($y = \text{rate}$ and $x = \text{concentration}$), first order ($y = t_{1/2}$ and $x = \text{concentration}$).