Study guide: Algebra2: Conic Equations and Features
Circle, parabola, ellipse and hyperbola equations encode geometric features through squared quantities and sign patterns. Standard forms locate centers, focal information, axes or vertices, with care to distinguish full lengths from half lengths.
Ideas to explain in your own words
- The squared-distance circle form (x−h)²+(y−k)²=r² encodes a center and nonnegative radius. The constants inside squared differences locate the center rather than directly supplying axis intercepts. The signs in the differences are opposite the center’s coordinate signs.
- The vertical parabola x²=4py consists of points equally distant from focus (0,p) and directrix y=−p. The coefficient four p determines the focal distance and opening direction. Divide the y coefficient by four to identify p.
- For an origin-centered ellipse with positive squared denominators, the larger denominator supplies the squared semi-major axis. Twice its positive root is the full major-axis length. The denominator is a squared half-length, not the full axis length.
- An origin-centered horizontal hyperbola has a positive x² term and a subtracted y² term. Setting y to zero locates the two vertices at plus and minus its horizontal semi-axis length. The positive squared term decides the transverse-axis direction.
Useful vocabulary
- conic
- A curve such as a circle, ellipse, parabola or hyperbola.
- focus
- A reference point used in a conic’s distance definition.
- semi-major axis
- Half the ellipse’s longest axis length.
- vertex
- A named turning or endpoint feature of a conic branch.
Watch your reasoning
- The signs in the differences are opposite the center’s coordinate signs. The right-side value is radius squared, so take its positive root rather than reporting that squared value as the radius.
- Divide the y coefficient by four to identify p. A focus is not the vertex: the unshifted vertex here is the origin, and positive p places the focus above it.
- The denominator is a squared half-length, not the full axis length. Identify which variable has the larger denominator to determine the major-axis direction before reporting a length.
A worked review example
Complete supplied guide examples: Read a Circle’s Center and Radius — Identify center and radius of (x−2)²+(y+3)²=4. Candidate answers: (A) Center (2,3); radius 2 (B) Center (2,−3); radius 2 (C) Center (2,−3); radius 4 (D) Center (−2,3); radius 2. Compare to (x−h)²+(y−k)²=r². Here h=2, k=−3, r=√4=2. The signs inside the parentheses are opposite the center coordinates. Read a Focus from a Standard Parabola — For x²=8y, find the focus. Candidate answers: (A) (0,2) (B) (0,8) (C) (0,−2) (D) (2,0). The standard vertical parabola form x²=4py has focus (0,p) and directrix y=−p. Thus p=8/4=2, and the focus is (0,2). Distinguish a Semi-Axis from a Full Axis — For x²/25+y²/4=1, what is the full major-axis length? Candidate answers: (A) 10 units (B) 25 units (C) 4 units (D) 5 units. The larger denominator is the square of the horizontal semi-major axis. Its half-length is 5; the full length is 2×5=10. Locate Vertices along the Positive-Term Axis — Find the vertices of x²/4−y²/9=1. Candidate answers: (A) (0,−3) and (0,3) (B) (−2,0) and (2,0) (C) (−3,0) and (3,0) (D) (−4,0) and (4,0). The positive x² term gives a horizontal transverse axis. Set y=0 to get x²/4=1, so x=±2.
Make a practice plan
- Study the four methods in the listed order; attempt each independent check before opening its explanation.
- Complete each lesson’s three changed-input questions and record the operation or reasoning step behind any correction.
- Take the four-question unit review, then solve the transfer task with written justifications.
- After correcting errors, attempt the separate unit exam and revisit the linked lesson for any unresolved method.