What does the graph of y = -x^2 look like?

The graph of \( y = -x^2 \) is a downward-facing parabola with its vertex at the origin.

In more detail, the equation \( y = -x^2 \) represents a quadratic function. The general form of a quadratic function is \( y = ax^2 + bx + c \). In this case, \( a = -1 \), \( b = 0 \), and \( c = 0 \). The negative coefficient of \( x^2 \) (which is -1) indicates that the parabola opens downwards, unlike \( y = x^2 \) which opens upwards.

The vertex of the parabola is the highest point on the graph, and for \( y = -x^2 \), this vertex is at the origin, which is the point (0, 0). This is because there are no additional terms to shift the graph horizontally or vertically.

The axis of symmetry for this parabola is the y-axis, or the line \( x = 0 \). This means that the graph is symmetrical on either side of the y-axis.

As you move away from the vertex along the x-axis, the value of \( y \) decreases because the square of any real number is positive, and multiplying by -1 makes it negative. For example, when \( x = 1 \), \( y = -1 \); when \( x = 2 \), \( y = -4 \); and so on. Similarly, for negative values of \( x \), the graph behaves the same way due to the symmetry.

The graph extends infinitely in the downward direction, getting wider as it goes further from the vertex. This shape is typical for any quadratic function with a negative leading coefficient.