Geometric Bites

A quadratic function and its turning point. Link to graph:  https://www.desmos.com/calculator/fktyfs12st A quadratic function is any function of the form f(x) = ax² + bx + c with a ≠ 0 . Its graph is a parabola, and every parabola has exactly one turning point (also called the vertex ). Completing the square is fundamental because it rewrites the quadratic as a shifted square , which makes the turning point immediately visible. Vertex form: the turning point is built in The vertex form of a quadratic is: f(x) = a(x − h)² + k This form is powerful because it exposes two facts at once: (x − h)² ≥ 0 for all real x (a square is never negative). (x − h)² = 0 happens exactly when x = h . So: If a > 0 , then a(x − h)² ≥ 0 , so the smallest possible value of f(x) is k , achieved at x = h (a minimum). If a < 0 , then a(x − h)² ≤ 0 , so the largest possible value of f(x) is k , achieved at x = h (a maximum). Therefore, in vertex form, the turning ...

Parabolas in sight: The Clifton Suspension Bridge, Bristol, United Kingdom. A quadratic function is a function whose graph is a parabola (a U-shaped curve). One of the most useful ways to write a quadratic is in vertex form , because it shows the parabola’s turning point immediately. 1) The vertex form A quadratic function in vertex form is written as: f(x) = a(x - h) 2 + k This form is especially helpful because the values h and k tell you the vertex directly. 2) The vertex (turning point) The vertex is the point where the parabola changes direction. In vertex form: Vertex = (h, k) If the parabola opens up , the vertex is the lowest point (a minimum). If the parabola opens down , the vertex is the highest point (a maximum). 3) What the number a does The number a controls two key things: the direction the parabola opens, and how wide or narrow it is. a > 0 means the parabola opens up (U-shape). a < 0 means the parabola opens dow...