Final answer:
The axis of symmetry, vertex, x-intercepts, y-intercept, and the number of real solutions describe a quadratic function's graph. These can be calculated using the quadratic equation's coefficients. The discriminant determines the number of real solutions.
Step-by-step explanation:
The axis of symmetry, the vertex, the x-intercept(s), the y-intercept, and the number of real number solutions are all characteristics that describe the graph of a quadratic function. The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves and it can be found using the formula x = -b/2a when the quadratic equation is in the form y = ax^2 + bx + c. The vertex is the point where the parabola reaches its maximum or minimum value, and it lies on the axis of symmetry. The vertex can also be found using the formula (-b/2a, f(-b/2a)) where f(x) is the quadratic function. The x-intercepts are the points where the graph crosses the x-axis, and these are found by solving the equation ax^2 + bx + c = 0. The y-intercept is the point where the graph crosses the y-axis, which occurs when x=0, so it's the value c in the quadratic equation. The number of real number solutions to the quadratic equation corresponds to the number of x-intercepts and is determined by the discriminant b^2 - 4ac. When the discriminant is positive, there are two real solutions; when it is zero, there is one; and when it is negative, there are no real solutions.