Quadratic Graphs

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

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

The vertex of a parabola is the highest or lowest point on the graph, depending on the direction it opens.

In vertex form $$y = a(x - h)^2 + k$$ , the vertex is the point (h, k).

The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves, given by the equation $$x = -\frac{b}{2a}$$ .

The 'a' value determines the direction of the parabola: if a > 0, it opens upwards; if a < 0, it opens downwards.

The discriminant, given by $$D = b^2 - 4ac$$ , determines the nature of the roots: if D > 0, there are two 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.

The maximum or minimum value occurs at the vertex. If 'a' is positive, the vertex is a minimum; if 'a' is negative, the vertex is a maximum.

The y-intercept is the point where the graph crosses the y-axis, found by evaluating the function at x = 0, i.e., $$c$$ in the standard form $$y = ax^2 + bx + c$$ .