You can easily identify the graph of the quadratic function $y = x^2 + 2x + 1$ as a parabola that opens upwards.

This equation represents a quadratic function, which is characterized by its parabolic shape. To better understand this specific parabola, we can rewrite the equation in vertex form by completing the square. By transforming $y = x^2 + 2x + 1$, we arrive at the expression:

This reveals that the vertex of the parabola is located at the point $(-1, 0)$.

The positive coefficient of $x^2$ confirms that the parabola opens upwards, which is a fundamental characteristic of its graph. Moreover, since the equation can be expressed as $y = (x + 1)^2$, it indicates that the graph has been shifted 1 unit to the left from the origin.

To plot the graph effectively, begin by marking the vertex at the point $(-1, 0)$. Next, select a few values of $x$ around the vertex, compute the corresponding $y$ values, and plot these points. For instance, when $x = 0$, we find $y = 1$; and when $x = -2$, we also have $y = 1$. These points will assist you in sketching the characteristic U-shaped curve of the parabola.

Keep in mind that parabolas exhibit symmetry about their vertex. This means that for every point $(x, y)$ located on one side of the vertex, there is a corresponding point $(-x - 2, y)$ on the opposite side. This property of symmetry can be very useful in ensuring that your graph is accurately represented.