Find the focus or directrix of a parabola

Learning Objectives Define the focus, directrix, vertex, and focal length (p) of a parabola. Identify the orientation (up, down, left, or right) of a parabola from its standard equation. Calculate the value of 'p' from the standard equation of a parabola. Determine the coordinates of the focus for a parabola with its vertex at the origin (0, 0). Write the equation of the directrix for a parabola with its vertex at the origin (0, 0). Calculate the coordinates of the focus for a parabola with a translated vertex at (h, k). Determine the equation of the directrix for a parabola with a translated vertex at (h, k). Ever wonder how a satellite dish focuses signals onto a single receiver? 📡 That's the magic of a parabola's focus point in action! This tutoria...

TermDefinitionExample ParabolaA set of all points in a plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix).The graph of the equation y = x² is a parabola. FocusThe fixed point from which all points on the parabola are equidistant. The parabola always curves around the focus.For the parabola x² = 8y, the focus is located at the point (0, 2). DirectrixThe fixed line from which all points on the parabola are equidistant. The parabola always curves away from the directrix.For the parabola x² = 8y, the directrix is the horizontal line y = -2. VertexThe turning point of the parabola. It is the midpoint on the axis of symmetry between the focus and the directrix.For the parabola (x - 3)² = 4(y + 1), the vertex is at the point (3, -1). Focal Length (p)The di...

Standard Equations for a Parabola with Vertex at (0, 0) Vertical: x² = 4py | Horizontal: y² = 4px Use these forms when the vertex is at the origin. If x is squared, the parabola is vertical (opens up for p > 0, down for p < 0). If y is squared, it is horizontal (opens right for p > 0, left for p < 0). The focus is at (0, p) for vertical and (p, 0) for horizontal. The directrix is y = -p for vertical and x = -p for horizontal. Standard Equations for a Parabola with Vertex at (h, k) Vertical: (x - h)² = 4p(y - k) | Horizontal: (y - k)² = 4p(x - h) Use these forms for any parabola. The vertex is (h, k). For a vertical parabola, the focus is at (h, k + p) and the directrix is y = k - p. For a horizontal parabola, the focus is at (h + p, k) and the directrix is x = h...