Graph parabolas

Learning Objectives Identify the vertex, focus, directrix, and axis of symmetry from the standard form equation of a parabola. Determine the direction of opening of a parabola based on its equation and the sign of the parameter 'p'. Convert the general form equation of a parabola into standard form by completing the square. Graph a parabola accurately by plotting its vertex, focus, and the endpoints of the latus rectum. Derive the standard form equation of a parabola given its key geometric features. Analyze the role of the parameter 'p' in determining the width and orientation of a parabola. Ever wonder how a satellite dish focuses signals onto a single receiver? 🛰️ The secret lies in the unique geometric property of the parabola! In this tutorial, we w...

TermDefinitionExample Parabola (Geometric Definition)The set of all points in a plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix).If the focus is at (0, 2) and the directrix is the line y = -2, the point (4, 2) is on the parabola because its distance to the focus (4 units) is equal to its distance to the directrix (4 units). VertexThe point on the parabola that lies on the axis of symmetry, midway between the focus and the directrix. It is the point where the parabola makes its sharpest turn.For the parabola (x-1)^2 = 8(y-2), the vertex is at (1, 2). FocusThe fixed point used to define a parabola. The parabola 'wraps around' the focus.For a parabola with vertex at (0,0) and equation x^2 = 12y, the focus is at (0, 3). DirectrixThe fixed li...

Standard Form of a Vertical Parabola (x-h)^2 = 4p(y-k) Use this form for parabolas that open upwards or downwards. The vertex is at (h, k). If p > 0, the parabola opens upwards. If p < 0, the parabola opens downwards. The focus is at (h, k+p) and the directrix is the line y = k-p. Standard Form of a Horizontal Parabola (y-k)^2 = 4p(x-h) Use this form for parabolas that open to the right or to the left. The vertex is at (h, k). If p > 0, the parabola opens to the right. If p < 0, the parabola opens to the left. The focus is at (h+p, k) and the directrix is the line x = h-p.