The graph of the function $y = -2^x$ represents a downward-sloping exponential curve that approaches zero but never intersects it.

To elaborate, the equation $y = -2^x$ exemplifies an exponential function characterized by a negative coefficient. This implies that as the value of $x$ increases, the value of $y$ decreases exponentially. The base of the exponent is 2, indicating that the function decays at a rate proportional to powers of 2. However, due to the negative sign preceding $2^x$, the graph is reflected across the x-axis relative to the graph of $y = 2^x$.

When $x = 0$, we find that $y = -2^0 = -1$. This gives us the y-intercept at the coordinate point $(0, -1)$. As $x$ takes on positive values, $y$ rapidly becomes more negative. For instance, when $x = 1$, $y = -2^1 = -2$, and when $x = 2$, $y = -2^2 = -4$. This demonstrates that the graph descends steeply as $x$ increases.

In contrast, when $x$ takes on negative values, $y$ approaches zero from the negative side. For example, when $x = -1$, we have $y = -2^{-1} = -\frac{1}{2}$, and when $x = -2$, $y = -2^{-2} = -\frac{1}{4}$. This behavior indicates that the graph gets increasingly closer to the x-axis without ever touching it, thereby establishing a horizontal asymptote at $y = 0$.

In summary, the graph of $y = -2^x$ is a steep, downward-sloping curve that begins at $y = -1$ when $x = 0$ and decreases rapidly as $x$ increases, while it approaches zero as $x$ decreases.