Unit 5 · Core unit

Conic Sections

Four curves that come from slicing a cone, unified by one idea: each is the set of points whose distances to fixed points or lines stand in a fixed relationship.

What a strong answer looks like

A strong answer here identifies the curve from its equation first, then reads the parameters. Naming the family before computing prevents applying the ellipse relationship to a hyperbola.

Practice this unit Course overview

Topics in this unit

Circles and Completing the Square

Know

A circle is the set of points at a fixed distance from a center. Standard form displays the center and radius directly, while general form hides them behind linear terms.

Apply

Complete the square on each variable to move from general to standard form, remembering to add the same constants to both sides of the equation.

Watch out

The center reads with the signs reversed: (x - 3) means the center coordinate is 3, not -3. Forgetting to balance the right side leaves the radius wrong.

Study move

Convert a general-form equation to standard form and then back again, confirming you land where you started.

Parabolas as a Focus and a Directrix

Know

A parabola is the set of points equidistant from a fixed point and a fixed line. Which variable is squared decides the axis, and the sign decides the direction of opening.

Apply

Match the equation to (y - k)² = 4p(x - h) or (x - h)² = 4p(y - k) to read p, then place the focus p units from the vertex and the directrix the same distance on the other side.

Watch out

Confusing 4p with p is the standard error. In y² = 12x the value of p is 3, not 12.

Study move

For each parabola, state the axis, the direction of opening, the focus, and the directrix before sketching anything.

Ellipses

Know

An ellipse is the set of points whose distances to two foci add to a constant. The larger denominator sits under the variable along the major axis.

Apply

Use c² = a² - b² to locate the foci, where a is the semi-major axis. The foci always lie on the major axis, inside the curve.

Watch out

The ellipse relationship subtracts. Using the hyperbola relationship here is the most common mistake in the unit, and it puts the foci outside the curve, which should be an immediate warning.

Study move

After finding c, check that it is smaller than a. If it is not, you used the wrong relationship.

Hyperbolas

Know

A hyperbola is the set of points whose distances to two foci differ by a constant. It has two branches and a pair of asymptotes that govern its far behavior.

Apply

Use c² = a² + b² for the foci. For a horizontal hyperbola centered at (h, k), the asymptotes are y - k = ±(b/a)(x - h); set h = k = 0 only when the center is the origin. The positive squared term identifies the transverse axis.

Watch out

For a hyperbola the larger denominator does not decide the orientation. The sign does, which is the reverse of the habit built on ellipses.

Study move

Sketch the central rectangle from a and b, draw its diagonals as the asymptotes, then hang the branches on them.

Identifying a Conic from General Form

Know

For a nondegenerate real conic with no xy term, one squared variable indicates a parabola, same-sign squared terms indicate an ellipse or circle, and opposite-sign squared terms indicate a hyperbola. Degenerate or imaginary cases require checking the completed-square form.

Apply

Classify tentatively from the squared terms, then complete the square to verify that a real, nondegenerate graph exists and to find its parameters. Equal same-sign coefficients narrow an ellipse to a circle.

Watch out

Linear terms only translate the curve. They never change which conic it is, so do not let them distract from the classification.

Study move

Given a mixed list of general-form equations, classify all of them before solving any, and say which feature decided each.

Emphasized in this unit

Connections and techniques that receive extra attention in this honors sequence.

• Conic sections as a geometric extension of function transformations and completing the square
• Completing the square as the single technique that unlocks all four curves
• The reflective properties that make parabolas and ellipses useful in engineering and astronomy
• Classifying from general form before attempting any parameters

Varies by course

Related topics some schools attach to this unit and others leave out. Covered on request rather than assumed.

Rotation of axes and the xy term. Rarely covered at this level. Most courses restrict conics to axes parallel to the coordinate axes.
Polar equations of conics. Appears in some courses that teach conics after polar coordinates, using eccentricity as the unifying parameter.

Mastery checklist

• Can convert a circle from general to standard form and read center and radius.
• Can find the focus and directrix of a parabola and say which way it opens.
• Can locate the foci of an ellipse and confirm they lie inside it.
• Can find the foci and asymptotes of a hyperbola.
• Can classify any conic from its general form before computing parameters.
• Can explain why a satellite dish and a whispering gallery use different conics.

Check yourself

• Why does the ellipse relationship subtract while the hyperbola relationship adds?
• What single feature of a general-form equation separates an ellipse from a hyperbola?
• Why does a parabola have one focus while an ellipse and a hyperbola each have two?

Modeling drill

A whispering gallery has an elliptical cross-section 40 feet wide and 16 feet from its lowest point to its highest point. Place the center at the origin, find the two foci, and explain why a person standing at one focus can hear someone speaking softly at the other.

conic sectioncircleparabolafocusdirectrixellipsemajor axisminor axishyperbolatransverse axisasymptoteeccentricitycompleting the squaregeneral form

Unit 4: Systems, Matrices, and Sequences

Unit 6: Limits and Continuity