Algebra II: Quadratic Functions: Challenge

Hard Level Guide Stretch thinking with multi-step problems, application questions, and deeper reasoning. Standard and Vertex Form Standard form is ax² + bx + c. Vertex form is a(x − h)² + k, where (h, k) is the vertex. The coefficient a controls whether the parabola opens up (a > 0) or down (a < 0) and how wide it is. Graphing Parabolas The axis of symmetry is x = h in vertex form, or x = −b/(2a) in standard form. The vertex is the maximum or minimum point. The y-intercept is c when x = 0. Plot symmetric points around the axis. Solving by Factoring Set ax² + bx + c equal to zero. Factor into two binomials whose product is zero. Use the zero product property: if ab = 0, then a = 0 or b = 0. Check solutions by substitution. The Quadratic Formula x = (−b ± √(b² − 4ac)) / (2a) solves any quadratic. The discriminant b² − 4ac tells how many real solutions exist: positive means two, zero means one, negative means none (in real numbers).