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...

1) n! (factorial) Meaning: tells you how many permutations (complete orderings) you can generate with n objects at your disposal. Definition: n! = n × (n−1) × (n−2) × … × 2 × 1, with 0! = 1. Example: 5! = 5×4×3×2×1 = 120. 2) n!/(n−r)! (permutations of r choices from n; order matters) Meaning: tells you how many ordered outcomes you can generate when you make r choices out of a collection of n objects, without reuse. How to see it: 1st choice: n options 2nd choice: (n−1) options 3rd choice: (n−2) options … rth choice: (n−r+1) options Multiply: n × (n−1) × … × (n−r+1) = n!/(n−r)!. Example (n=5, r=2): 5P2 = 5!/(5−2)! = 5!/3! = (5×4×3×2×1)/(3×2×1) = 5×4 = 20. 3) (n!/(n−r)!)/r! = n!/((n−r)! r!) (combinations; order neglected) Meaning: tells you how many selections you can make when choosing r objects from n, where order does not matter. ...

Geometric Bites has always been about clear diagrams and full derivations that help you see the structure of mathematics. Over time it has grown into a rich, freely accessible library of explanations, proofs and visual ideas. Geometry Insights is the premium companion to that work: a dedicated article site focused on GCSE, A-Level and Further Pure mathematics, written from first principles with carefully engineered diagrams and a long-term, structured archive in mind. Visit Geometry Insights: https://geometryinsights.wordpress.com What Makes Geometry Insights Different? Where Geometric Bites offers free posts and full derivations, Geometry Insights is a curated, subscription-only library. Each article is built to answer a deeper question: not just “how do I use this formula?” but “why does this formula exist at all?” Premium-only articles that go in depth on GCSE, A-Level and Further Pure topics. First-principles derivations that start from definitions and basic f...

The angle–difference identities are: sin(α − β) = sinα cosβ − cosα sinβ cos(α − β) = cosα cosβ + sinα sinβ They can be seen geometrically using a rectangle OZYX with a few right-angled triangles inside it. All side lengths can be written in terms of sinα, cosα, sinβ and cosβ. First, draw a right triangle OPQ with hypotenuse OQ = 1 and angle β at O. By definition: OP = cosβ (horizontal side), QP = sinβ (vertical side). Next, use OP as the hypotenuse of another right triangle OPZ. The right angle is at Z, and the angle at P is α. The hypotenuse is OP = cosβ, so: OZ = sinα cosβ, ZP = cosα cosβ. In a similar way, use QP as the hypotenuse of a right triangle PQY. The right angle is at Y, and the angle at Q is α. The hypotenuse is QP = sinβ, so: QY = cosα sinβ, PY = sinα sinβ. Drop a vertical line from Y to the base at Z, and a horizontal line from Y to the left side at X. This makes OZYX a rectangle with: base OZ, height ZY. On the...

Consider a cyclic quadrilateral: all four of its vertices lie on a circle. Join the centre of the circle to each vertex. This creates four isosceles triangles, each made from two radii and one side of the quadrilateral. Label the angles of the quadrilateral at the circumference by w, x, y and z. In each isosceles triangle, the angles at the base are equal, so the angle at the centre is: π − 2w, π − 2x, π − 2y, π − 2z. These four central angles meet at a point, so together they make one full turn: (π − 2w) + (π − 2x) + (π − 2y) + (π − 2z) = 2π. Rearranging gives: −2w − 2x − 2y − 2z + 4π = 2π so 2w + 2x + 2y + 2z = 2π and therefore w + x + y + z = π. From this, the opposite-angle relations in the quadrilateral follow directly: w + z = π − (x + y), z + y = π − (w + x). So in any cyclic quadrilateral, each pair of opposite angles adds up to π radians .