What equation represents a concave-down graph?

A concave-down graph is characterized by its shape resembling an upside-down cup or a frown. This occurs when the second derivative of the function is negative across the domain of interest, indicating that the slope of the tangent line decreases as we move along the graph.

The equation y = -x^2 is a quadratic function where the coefficient of x^2 is negative. This negative leading coefficient means the parabola opens downward, creating a concave-down shape. As you move away from the vertex, the y-values decrease, which is a clear indication of the graph's concavity being downwards.

In contrast, y = x^2 represents a concave-up graph, as it opens upward due to the positive leading coefficient. The linear equations, such as y = 3x and y = x + 1, yield straight lines, which do not exhibit any concavity. Straight lines have constant slopes and, therefore, do not possess the property of being concave up or down.