Overview
This lecture covers the equations and properties of parabolas, including how to graph them, find key features (vertex, focus, directrix, latus rectum), match equations to graphs, and convert to standard form.
Parabola Basics
- Parabolas can be represented as ( y^2 = 4px ) or ( x^2 = 4py ) with the vertex at the origin.
- The focus is ( p ) units from the vertex; the directrix is also ( p ) units from the vertex in the opposite direction.
- The directrix for ( y^2 = 4px ) is ( x = -p ); for ( x^2 = 4py ), it's ( y = -p ).
- A positive ( p ) determines the direction the parabola opens (right/up); a negative ( p ) reverses it (left/down).
- The latus rectum (focal diameter) is a line segment through the focus and has length ( 4p ).
Graphing Parabolas & Key Features
- Identify ( p ) by dividing the coefficient of the non-squared variable by 4.
- The vertex is at the origin or at ( (h, k) ) if shifted.
- The focus is at ( (h+p, k) ) for horizontal, ( (h, k+p) ) for vertical parabolas.
- The directrix is ( x = h-p ) (horizontal) or ( y = k-p ) (vertical).
- For a parabola not at the origin, use ( (y-k)^2 = 4p(x-h) ) or ( (x-h)^2 = 4p(y-k) ).
Equation-Graph Orientation Matching
- ( x^2 = 4py ): opens up if ( p>0 ), down if ( p<0 ).
- ( y^2 = 4px ): opens right if ( p>0 ), left if ( p<0 ).
- Negative leading coefficients flip the opening direction.
Standard Form & Completing the Square
- For equations not in standard form, complete the square to rewrite as ( (x-h)^2 = 4p(y-k) ) or ( (y-k)^2 = 4p(x-h) ).
- Identify vertex ( (h, k) ) directly from the standard form.
Domain & Range
- For horizontal parabolas: range is all real ( y ), domain starts at vertex ( x ) and extends to infinity.
- For vertical parabolas: domain is all real ( x ), range depends on opening direction (up: ( [k,\infty) ), down: ( (-\infty,k] )).
Key Terms & Definitions
- Parabola — The set of all points equidistant from a point (focus) and a line (directrix).
- Vertex — The turning point of the parabola.
- Focus — The point inside the parabola used to define it.
- Directrix — The line outside the parabola used to define it.
- Latus Rectum — The line segment through the focus, perpendicular to the axis, length ( 4p ).
- Standard form — ( (x-h)^2 = 4p(y-k) ) or ( (y-k)^2 = 4p(x-h) ).
Action Items / Next Steps
- Practice identifying ( p ), vertex, focus, and directrix for given equations.
- Convert general form equations to standard form using completing the square.
- Graph several types of parabolas and label all features.
- Review formulas for shifted parabolas and memorize key standard forms.