How are the real solutions of a quadratic equation related to the graph of the quadratic function?
The real solutions of a quadratic equation are related to the graph of the quadratic function in the following ways:
1. x-intercepts: The x-values of the real solutions of the quadratic equation correspond to the x-intercepts of the quadratic function's graph. These are the points where the function crosses the x-axis.
2. Vertex: The vertex of the quadratic function represents the minimum or maximum point of the parabolic curve. The x-coordinate of the vertex can be found using the formula x = -b/2a, where the quadratic equation is in the form ax^2 + bx + c = 0.
a. If the quadratic equation has two real solutions, the vertex lies exactly in the middle between these two solutions on the x-axis.
b. If the quadratic equation has one real solution (a double root), the vertex coincides with this solution on the x-axis.
3. Discriminant: The discriminant of the quadratic equation, which is given by the expression b^2 - 4ac, provides information about the nature of the solutions.
a. If the discriminant is positive (b^2 - 4ac > 0), the quadratic equation has two distinct real solutions. On the graph, this means the parabola intersects the x-axis at two different points.
b. If the discriminant is zero (b^2 - 4ac = 0), the quadratic equation has one real solution (a double root). On the graph, this means the parabola touches the x-axis at exactly one point.
c. If the discriminant is negative (b^2 - 4ac < 0), the quadratic equation has no real solutions (only complex solutions). On the graph, this means the parabola does not intersect the x-axis, i.e., it lies entirely above or below the x-axis.
The real solutions of a quadratic equation are related to the graph of the corresponding quadratic function in the following way:
1. The number of real solutions: The number of real solutions of the quadratic equation (zero, one, or two) corresponds to the number of times the graph of the quadratic function intersects or touches the x-axis.
2. x-intercepts: If the quadratic equation has real solutions, they represent the x-coordinates of the points where the graph of the quadratic function intersects the x-axis. These points are also known as the x-intercepts or roots of the quadratic function.
3. Vertex: The vertex of the parabolic graph of the quadratic function serves as the mid-point between the x-intercepts (real solutions) if they exist. The x-coordinate of the vertex can be found using the formula x = -b / (2a), where a and b are the coefficients of the quadratic equation in the standard form.
4. Direction: The direction of the graph's opening depends on the leading coefficient (a) of the quadratic equation. If a is positive, the graph opens upward, and if a is negative, the graph opens downward.
By understanding these relationships, you can use the quadratic equation and its solutions to determine various characteristics of the graph of the quadratic function.