Question

which statement best describes the relationship between a quadratic equations solutions and its graph?

the solution is the y-intercept of the parabola.

the solutions are the slopes of each side of the parabola.

the solutions are the x-intercepts of the parabola.

the solution is either the maximum or minimum of the parabola.

Explanation:

To determine the correct statement, we analyze each option:

• The y - intercept of a parabola (from a quadratic function \(y = ax^{2}+bx + c\)) occurs when \(x = 0\), and it is the value of \(c\). The solutions of the quadratic equation \(ax^{2}+bx + c=0\) are not the y - intercept.
• A parabola is a curve, and the concept of slope (which is for straight lines) does not apply to the "sides" of a parabola in the way described. The solutions of the quadratic equation are not slopes.
• For a quadratic equation \(ax^{2}+bx + c = 0\), the solutions (also called roots) are the values of \(x\) for which \(y=0\) in the quadratic function \(y=ax^{2}+bx + c\). The \(x\) - intercepts of the graph of the quadratic function (a parabola) are the points where \(y = 0\), so the solutions of the quadratic equation are the \(x\) - intercepts of the parabola.
• The maximum or minimum of a parabola (the vertex) has a \(y\) - value (and an \(x\) - value for the vertex's location), but it is not the solution of the quadratic equation \(ax^{2}+bx + c = 0\) (unless the vertex is also an \(x\) - intercept, which is a special case). The solution of the quadratic equation is not defined as the maximum or minimum in general.

Answer:

C. The solutions are the x - intercepts of the parabola.