Unit 7 · Topic 4.6 · Optional
Conic Sections
Recognize circles, parabolas, ellipses, and hyperbolas from their equations, then extract centers or vertices, symmetry, dimensions, and asymptotes from standard form.
Learning Goals
- Classify an axis-aligned conic from its squared terms and signs.
- Interpret translations through \((x-h)\) and \((y-k)\).
- Graph horizontal and vertical parabolas, ellipses, and hyperbolas.
- Determine vertices, radii, axes of symmetry, and asymptotes.
- Rewrite selected general-form equations by completing the square.
1. Four Related Curve Families
A conic section can be formed by intersecting a plane with a double cone. In the coordinate plane, its equation relates x and y quadratically.
| Conic | Geometric idea | Graph structure |
|---|---|---|
| Circle | Fixed distance from one center | Closed and equally wide in every direction |
| Parabola | Equal distance from a focus and directrix | One open branch |
| Ellipse | Constant sum of distances from two foci | Closed oval |
| Hyperbola | Constant absolute difference of distances from two foci | Two open branches |
2. Classify from Squared Terms
| Squared-variable pattern | Likely nondegenerate conic |
|---|---|
| Only x or only y is squared | Parabola |
| Both squared terms have the same sign and equal scaled coefficients | Circle |
| Both squared terms have the same sign but unequal scaled coefficients | Ellipse |
| Squared terms have opposite signs | Hyperbola |
This lesson considers equations with horizontal or vertical symmetry and no \(xy\)-term.
3. Read Translations Carefully
The expressions \((x-h)\) and \((y-k)\) place the center or vertex at \((h,k)\).
so \(x+3\) corresponds to a horizontal coordinate of \(-3\), not \(3\). Signs inside grouped squares appear opposite the graph's coordinate.
4. Compare the Three Main Shapes
5. Standard Forms of a Parabola
| Equation | Axis | Opening |
|---|---|---|
| \(y-k=a(x-h)^2\) | Vertical, \(x=h\) | Up if \(a>0\), down if \(a<0\) |
| \(x-h=a(y-k)^2\) | Horizontal, \(y=k\) | Right if \(a>0\), left if \(a<0\) |
Both forms have vertex \((h,k)\). The variable that is not squared identifies the direction in which the parabola opens.
6. Parabola Example
Analyze
The vertex is \((-1,2)\), the axis of symmetry is \(x=-1\), and the negative coefficient makes the parabola open downward. In the equivalent focus form \((x+1)^2=-4(y-2)\), \(4p=-4\), so the focus is \((-1,1)\) and the directrix is \(y=3\).
7. Standard Form of an Ellipse
The center is \((h,k)\), the horizontal radius is \(|a|\), and the vertical radius is \(|b|\). The larger denominator identifies the major-axis direction. When \(a^2=b^2\), the ellipse is a circle.
8. Ellipse Example
For
the center is \((0,0)\), horizontal radius 3, and vertical radius 2. The vertices are \((\pm3,0)\), and the co-vertices are \((0,\pm2)\).
9. Translated Ellipse Example
The center is \((2,-1)\). Since \(25>9\), the major axis is horizontal. The vertices are \((-3,-1)\) and \((7,-1)\); the co-vertices are \((2,-4)\) and \((2,2)\).
For additional geometric information, \(c^2=25-9=16\), so the foci are \((-2,-1)\) and \((6,-1)\).
10. A Circle Is a Special Ellipse
Equal horizontal and vertical radii produce a circle. Unlike a noncircular ellipse, a circle has infinitely many axes of symmetry through its center.
11. Standard Forms of a Hyperbola
| Equation | Opening | Asymptotes |
|---|---|---|
| \(\dfrac{(x-h)^2}{a^2}-\dfrac{(y-k)^2}{b^2}=1\) | Left and right | \(y-k=\pm\dfrac{b}{a}(x-h)\) |
| \(\dfrac{(y-k)^2}{a^2}-\dfrac{(x-h)^2}{b^2}=1\) | Up and down | \(y-k=\pm\dfrac{a}{b}(x-h)\) |
The positive squared term identifies the opening direction. The center \((h,k)\) is the intersection of the asymptotes and lies between the branches, not on the graph.
12. Horizontal Hyperbola Example
The center is \((-1,2)\), \(a=3\), and \(b=2\). The graph opens left and right with vertices
and asymptotes
13. Vertical Hyperbola Example
The center is \((2,3)\), \(a=4\), and \(b=3\). The graph opens up and down, has vertices \((2,7)\) and \((2,-1)\), and approaches
14. Rewrite General Form by Completing Squares
Classify \(x^2+4y^2-6x+8y-3=0\):
The graph is an ellipse centered at \((3,-1)\) with horizontal radius 4 and vertical radius 2.
15. AP Workflow and Common Errors
- Group x-terms, y-terms, and constants.
- Inspect the number and signs of squared terms.
- Complete squares when the equation is not in standard form.
- Normalize the right side to 1 for ellipses and hyperbolas.
- Identify the center or vertex and all symmetry axes.
- Read radii, opening direction, vertices, and asymptotes from the correct form.
- Plot defining features before sketching the curve.
- Do not use denominator values as lengths; take square roots.
- Do not assign the larger ellipse denominator automatically to x.
- Do not assume the center of a hyperbola lies on the graph.
- Do not reverse the asymptote ratio \(b/a\) for a horizontal hyperbola.
- Do not lose the inside-sign reversal in \((x-h)\) and \((y-k)\).
Key Takeaways
- Recognize circles, parabolas, ellipses, and hyperbolas from equations and graph features.
- Core relationship: \(\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1\)
- Error check: Do not take a denominator itself as a semiaxis length; use its square root.
- Classify \(4x^2+9y^2=36\) and identify its horizontal and vertical radii.
- Describe the vertex, axis, and opening direction of \(x+2=\frac13(y-1)^2\).
- Find the center, vertices, and co-vertices of \(\frac{(x+1)^2}{4}+\frac{(y-3)^2}{25}=1\).
- Find the vertices and asymptotes of \(\frac{(x-2)^2}{16}-\frac{(y+1)^2}{9}=1\).
- Complete the square to classify \(x^2+y^2-4x+6y-12=0\).