The graph of the function $y = e^x$ is characterized by an upward-sloping curve that becomes increasingly steep as $x$ increases.

The function $y = e^x$ represents an exponential function, where $e$ is a mathematical constant approximately equal to $2.71828$. This constant is known as Euler’s number. Notably, the graph of $y = e^x$ passes through the point $(0, 1)$ since any number raised to the power of $0$ equals $1$. As $x$ increases, the value of $y$ grows rapidly, resulting in a steep ascent of the graph.

For negative values of $x$, the graph approaches the x-axis but never intersects it. This behavior occurs because $e$ raised to a negative exponent produces a fraction that decreases continuously, remaining positive but never reaching zero. Consequently, the graph has a horizontal asymptote at $y = 0$.

The graph remains entirely above the x-axis, indicating that $y$ is always positive. This reinforces the fact that $e^x$ is greater than zero for all real numbers $x$. Additionally, the curve is smooth and continuous, exhibiting no breaks or gaps.

In summary, the graph of $y = e^x$ begins just above the x-axis on the left side, passes through the point $(0, 1)$, and then rises steeply as $x$ increases. This graph is fundamental in mathematics, particularly in discussions related to growth and decay, and effectively illustrates the concept of exponential growth.