Overview
This lecture covers how to identify, graph, and write standard forms for conic sections: circles, ellipses, hyperbolas, and parabolas, including distinguishing features and key formulas.
Circles
- Standard form: (x - h)² + (y - k)² = r², center at (h, k), radius r.
- If h and k are missing, the center is at (0, 0).
- To graph: plot the center, move r units in all directions, and draw the circle.
Ellipses
- Standard form: (x - h)²/a² + (y - k)²/b² = 1, a > b.
- Center: (h, k); major axis length is 2a, minor axis length is 2b.
- Major axis is along variable with larger denominator (a²).
- Vertices: endpoints of major axis; coordinates involve adding/subtracting a or b to center.
- Foci: c² = a² - b²; foci lie along major axis.
- X-intercepts: plug y = 0; Y-intercepts: plug x = 0.
Hyperbolas
- Standard forms: (x - h)²/a² - (y - k)²/b² = 1 (opens left/right), (y - k)²/a² - (x - h)²/b² = 1 (opens up/down).
- Center: (h, k); a: distance from center to vertex.
- Vertices: h ± a, k or h, k ± a according to the form.
- Foci: c² = a² + b²; foci farther from center than vertices.
- Asymptotes: y - k = ±(b/a)(x - h) for horizontal transverse axis; switch a/b for vertical.
- Graph opens along the variable that comes first.
Parabolas
- Standard forms: (y - k)² = 4p(x - h) (horizontal axis), (x - h)² = 4p(y - k) (vertical axis).
- Vertex at (h, k); focus is p units from vertex (direction depends on sign of p).
- Directrix at x = h - p or y = k - p (opposite side of focus).
- Opens right/left if y is squared; up/down if x is squared.
- To graph, make a table of points or use focus/directrix.
Identifying Conic Sections from Equations
- Circle: x² and y², same coefficients and sign.
- Ellipse: x² and y² positive, different coefficients.
- Hyperbola: x² and y², one positive, one negative.
- Parabola: only one squared term.
Converting to Standard Form (Completing the Square)
- Group x and y terms; factor as needed.
- Complete the square for both x and y.
- For ellipses/hyperbolas, set right side to 1 (divide both sides).
- For parabolas, arrange as one squared term = linear term, complete the square for variable squared.
Key Terms & Definitions
- Center — the fixed point (h, k) defining a circle, ellipse, or hyperbola.
- Vertex — endpoint of major axis (ellipse) or point where parabola/hyperbola changes direction.
- Focus (Foci) — fixed point(s) used in conic definitions; ellipses/hyperbolas have two.
- Directrix — fixed line used in parabola definition.
- Major axis — longest axis of an ellipse.
- Minor axis — shortest axis of an ellipse.
- Transverse axis — line through the vertices of a hyperbola.
- Asymptote — line that a hyperbola approaches but never touches.
- Radius — distance from center to circle’s edge.
- Standard form — equation form making key features easily readable.
Action Items / Next Steps
- Practice converting general conic equations to standard form using completing the square.
- Graph multiple examples of each conic section for familiarity.
- Memorize standard forms and relationships (such as how to find foci and axes lengths).