Explain how the graph of a quadratic function relates to the solutions of the related quadratic equation.

The graph of a quadratic function represents a parabola, which is a U-shaped curve. The quadratic equation is the equation of the form $ax^2 + bx + c = 0$, where a, b, and c are constants and x is the variable. The solutions of the quadratic equation are the values of x where the graph of the quadratic function crosses the x-axis.

If the parabola intersects the x-axis at two distinct points, then the quadratic equation has two real solutions. These solutions are the x-coordinates of the points where the parabola intersects the x-axis.

If the parabola is tangent to the x-axis at one point, then the quadratic equation has one real solution. This solution is the x-coordinate of the point where the parabola touches the x-axis.

If the parabola does not intersect or touch the x-axis at all, then the quadratic equation has no real solutions.

In summary, the graph of a quadratic function provides a visual representation of the solutions of the related quadratic equation, showing how many solutions exist and where they are located on the coordinate plane.