Quadratic Word Problem

A quadratic function is a polynomial function of degree 2, typically in the form of $$f(x) = ax^2 + bx + c$$ , where a, b, and c are constants and a ≠ 0.

The vertex of a quadratic function is the highest or lowest point on the graph, depending on the direction of the parabola. It can be found using the formula $$x = -\frac{b}{2a}$$ .

The maximum or minimum value of a quadratic function occurs at the vertex. Substitute the x-coordinate of the vertex back into the function to find the corresponding y-coordinate.

The 'a' value in a quadratic function determines the direction of the parabola. If 'a' is positive, the parabola opens upwards; if 'a' is negative, it opens downwards.

The standard form of a quadratic equation is $$ax^2 + bx + c = 0$$ .

The factored form of a quadratic function is $$y = a(x - r_1)(x - r_2)$$ , where r_1 and r_2 are the roots of the equation.

The roots of a quadratic equation can be found using the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ .

The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves. It can be found using the formula $$x = -\frac{b}{2a}$$ .

The discriminant, given by $$D = b^2 - 4ac$$ , determines the nature of the roots: if D > 0, there are two distinct real roots; if D = 0, there is one real root; if D < 0, there are no real roots.

To convert from standard form $$y = ax^2 + bx + c$$ to vertex form $$y = a(x - h)^2 + k$$ , complete the square.