Writing Quadratics in Vertex and Intercept Form

The vertex form of a quadratic equation is given by $$y = a(x - h)^2 + k$$ , where (h, k) is the vertex of the parabola.

To find the vertex of a parabola given in standard form $$y = ax^2 + bx + c$$ , use the formula $$h = -\frac{b}{2a}$$ and then substitute h back into the equation to find k.

The x-intercepts of a quadratic function are the points where the graph intersects the x-axis, found by solving the equation $$ax^2 + bx + c = 0$$ .

The 'a' value in a quadratic equation determines the direction of the parabola (upward if a > 0, downward if a < 0) and affects the width of the parabola.

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

The intercept form of a quadratic equation is given by $$y = a(x - p)(x - q)$$ , where p and q are the x-intercepts of the parabola.

The y-intercept of a quadratic function is found by evaluating the function at x = 0, i.e., $$y = a(0)^2 + b(0) + c = c$$ .

The domain of a quadratic function is all real numbers, represented as $$(-\infty, \infty)$$ .

The range of a quadratic function in vertex form $$y = a(x - h)^2 + k$$ depends on the value of 'a': if a > 0, the range is $$[k, \infty)$$ ; if a < 0, the range is $$(-\infty, k]$$ .

The direction of a parabola is determined by the sign of 'a' in the equation: if 'a' is positive, the parabola opens upward; if 'a' is negative, it opens downward.