What will the graph of a quadratic function passing through the origin (0,0) look like?

A quadratic function can be expressed in the general form ( f(x) = ax^2 + bx + c ). If the function passes through the origin, the point (0,0) means that when ( x = 0 ), ( f(x) = 0 ). This indicates that the constant term ( c ) is equal to 0, so the function simplifies to ( f(x) = ax^2 + bx ).

The key feature of this function is the value of ( a ), which determines the direction in which the parabola opens. If ( a > 0 ), the graph opens upwards, and if ( a < 0 ), the graph opens downwards. This means that the shape of the graph can vary significantly depending on the value of ( a ). Importantly, since the function must intersect the origin, it will cross the x-axis at that point ( (0,0) ).

This characteristic demonstrates that the graph of a quadratic function passing through the origin can indeed open in either direction (upwards or downwards), contradicting options suggesting limitations on its shape or intersection with the x-axis. Therefore, the correct choice aptly captures the essential properties of quadratic functions that pass through