Find properties of a parabola from equations in general form

Learning Objectives Identify whether a parabola opens vertically or horizontally from its general form. Convert the general form of a parabola's equation to its standard (vertex) form using the completing the square method. Determine the coordinates of the vertex from the standard form. Calculate the value of 'p' and use it to find the coordinates of the focus. Determine the equation of the directrix and the axis of symmetry. State the length of the latus rectum. Sketch a graph of the parabola using its key properties. Ever wondered how a satellite dish focuses signals to a single point? 📡 It's all about finding the 'focus' of a parabola, a skill you're about to master! This tutorial will teach you how to take a complex-looking paraboli...

TermDefinitionExample General Form of a ParabolaAn equation where the terms are not grouped to reveal the parabola's properties. It takes the form Ax² + Dx + Ey + F = 0 (for vertical parabolas) or Cy² + Dx + Ey + F = 0 (for horizontal parabolas).x² - 6x - 8y + 1 = 0 is the general form of a vertical parabola. Standard (Vertex) FormThe form of the equation that clearly reveals the vertex (h, k). It is written as (x - h)² = 4p(y - k) for vertical parabolas or (y - k)² = 4p(x - h) for horizontal parabolas.(x - 3)² = 8(y + 1) is the standard form, showing a vertex at (3, -1). Vertex (h, k)The turning point of the parabola; the minimum point if it opens up, or the maximum point if it opens down.For the parabola (y + 2)² = -12(x - 5), the vertex is (5, -2). FocusA fixed point inside the pa...

Vertical Parabola Conversion General Form: Ax² + Dx + Ey + F = 0 → Standard Form: (x - h)² = 4p(y - k) Use this when the equation contains an x² term. The parabola opens up if p > 0 and down if p < 0. The key is to isolate the x-terms and complete the square. Horizontal Parabola Conversion General Form: Cy² + Dx + Ey + F = 0 → Standard Form: (y - k)² = 4p(x - h) Use this when the equation contains a y² term. The parabola opens right if p > 0 and left if p < 0. The key is to isolate the y-terms and complete the square. Completing the Square For an expression like z² + bz, add (b/2)² to create a perfect square trinomial: z² + bz + (b/2)² = (z + b/2)². This is the essential algebraic technique used to transform the general form into the standard form. Re...