Write equations of parabolas in vertex form

Learning Objectives Identify the vertex (h, k) and orientation of a parabola from given information. Write the equation of a vertical parabola in vertex form given the vertex and another point on the curve. Write the equation of a horizontal parabola in vertex form given the vertex and another point on the curve. Derive the equation of a parabola in vertex form given its focus and directrix. Convert the equation of a parabola from general form to vertex form by completing the square. Model and solve real-world problems using parabolic equations in vertex form. Ever wondered about the perfect arc of a basketball shot or the shape of a satellite dish focusing signals from space? 🛰️ Those are all parabolas, and their secrets are unlocked by one powerful equation form. This tut...

TermDefinitionExample ParabolaAs a conic section, a parabola is the 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 quadratic function y = x^2 is a simple parabola with its vertex at the origin. Vertex (h, k)The vertex is the turning point of the parabola. It is the point where the parabola reaches its maximum or minimum value and where the axis of symmetry intersects the curve.For the parabola y = (x - 3)^2 + 5, the vertex is at the point (3, 5). Axis of SymmetryA line that passes through the vertex and divides the parabola into two perfectly mirrored halves.For a vertical parabola with vertex (h, k), the axis of symmetry is the vertical line x = h. For a horizontal parabola, it is the horizontal line y =...

Vertex Form for a Vertical Parabola y = a(x - h)^2 + k Use this form for parabolas that open upwards (if a > 0) or downwards (if a < 0). The vertex is (h, k) and the axis of symmetry is the vertical line x = h. Vertex Form for a Horizontal Parabola x = a(y - k)^2 + h Use this form for parabolas that open to the right (if a > 0) or to the left (if a < 0). The vertex is (h, k) and the axis of symmetry is the horizontal line y = k. Focus-Directrix Relationship a = \frac{1}{4p} This crucial formula connects the parameter 'a' to 'p', which is the directed distance from the vertex to the focus. This is essential when deriving an equation from the focus and directrix.