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

Chapter introduction

Understand the idea before memorizing the rule

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.

In Precalculus Honors, success means moving among words, symbols, tables, and graphs while preserving the meaning of each quantity. Use the lessons below as connected notes: explain the concept, carry out the procedure, test the result, and interpret it in context.

Detailed study notes

Core lessons, reasoning, and common errors

Lesson 1

Circles and Completing the Square

Concept explanation

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.

How to apply it

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.

Common mistake

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.

Guided study move

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

Lesson 2

Parabolas as a Focus and a Directrix

Concept explanation

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.

How to apply it

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.

Common mistake

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

Guided study move

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

Lesson 3

Ellipses

Concept explanation

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.

How to apply it

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.

Common mistake

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.

Guided study move

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

Lesson 4

Hyperbolas

Concept explanation

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.

How to apply it

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.

Common mistake

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

Guided study move

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

Lesson 5

Identifying a Conic from General Form

Concept explanation

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.

How to apply it

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.

Common mistake

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

Guided study move

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

Formula and visual reference

Connect the symbols to the picture

Circle: (x − h)² + (y − k)² = r²

Ellipse: (x − h)²/a² + (y − k)²/b² = 1

Hyperbola: (x − h)²/a² − (y − k)²/b² = 1

Parabola: (x − h)² = 4p(y − k)

Concept graph

The larger denominator identifies the major-axis direction.

Complete solution

Follow the reasoning, not just the arithmetic

Cover each next step and predict it before reading on. Then redo the example with one number changed to confirm that the method—not the answer—stuck.

Worked example

Read an ellipse from standard form

Problem

Analyze (x − 2)²/25 + (y + 1)²/9 = 1.

1The center is (2, −1). The larger denominator 25 lies under x, so the major axis is horizontal.
2a = 5 and b = 3, giving vertices (−3, −1) and (7, −1).
3c² = a² − b² = 16, so c = 4 and the foci are (−2, −1) and (6, −1).

Answer and interpretation

Center (2, −1), horizontal vertices (−3, −1) and (7, −1), foci (−2, −1) and (6, −1).

Chapter synthesis

Ideas to connect

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

Modeling lab

Transfer the chapter to a new setting

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.

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

conic sectioncircleparabolafocusdirectrixellipsemajor axisminor axishyperbolatransverse axisasymptoteeccentricitycompleting the squaregeneral form

Unit 4: Systems, Matrices, and Sequences

Unit 6: Limits and Continuity