Quadratic Functions and Their Graphs

Every quadratic function can be written in three equivalent forms. Each reveals different information: | Form | Equation | Reveals | |---|---|---| | Standard | f(x) = ax^2 + bx + c | y-intercept (c), direction (a) | | Vertex | f(x) = a(x - h)^2 + k | Vertex (h, k), direction (a) | | Factored | f(x) = a(x - r)(x - s) | x-intercepts (r and s) | All three describe the same parabola — just differen…

| Feature | How to find it | |---|---| | Direction | a > 0: opens up (∪, minimum). a < 0: opens down (∩, maximum) | | Vertex | From vertex form: (h, k). From standard: h = -frac{b}{2a}, k = f(h) | | Axis of symmetry | x = h (vertical line through vertex) | | Y-intercept | f(0) = c (in standard form) | | X-intercepts | Solve f(x) = 0 (factor, formula, or Desmos) | | Width | |a| large → narrow. |a|…

Example: For f(x) = -2(x - 3)^2 + 8, identify: vertex, direction, axis of symmetry, max/min.