Pre-Calculus - Week 7

Conic Sections: Circles and Parabolas

Conic sections are the curves you get by slicing a cone with a plane: circles, parabolas, ellipses, and hyperbolas. They appear in planetary orbits (ellipses), satellite dishes (parabolas), and telescope design (hyperbolas). This week focuses on circles and parabolas — both defined geometrically and analyzed algebraically.

Learning Objectives

- Write and identify the standard form of a circle equation. - Complete the square to convert general form to standard form. - Find the center and radius of a circle. - Write equations of parabolas in standard form with vertex at the origin. - Write equations of parabolas in vertex form with any vertex. - Identify the vertex, focus, directrix, and axis of symmetry of a parabola. - Graph circles and parabolas and label key features.

1. The Circle: Definition and Standard Form

A circle is the set of all points equidistant from a fixed center. Standard form: (x - h)^2 + (y - k)^2 = r^2 Center: (h, k). Radius: r. Example 1: (x - 3)^2 + (y + 1)^2 = 25. Center: (3, -1). Radius: 5. Example 2: x^2 + y^2 = 16. Center: (0, 0). Radius: 4. Example 3: Write the equation of the circle with center (-2, 5) and radius 3. (x + 2)^2 + (y - 5)^2 = 9.

2. General Form of a Circle

General form: x^2 + y^2 + Dx + Ey + F = 0. To identify the circle, complete the square to convert to standard form. Example: Convert x^2 + y^2 - 4x + 6y - 3 = 0. Group: (x^2 - 4x) + (y^2 + 6y) = 3. Complete the square: (x^2 - 4x + 4) + (y^2 + 6y + 9) = 3 + 4 + 9. (x - 2)^2 + (y + 3)^2 = 16. Center: (2, -3). Radius: 4.

3. Completing the Square: The Technique

To complete the square for x^2 + bx: 1. Take half of b: b/2. 2. Square it: (b/2)^2. 3. Add this to both sides of the equation. The expression x^2 + bx + (b/2)^2 = (x + b/2)^2. For ax^2 + bx first factor out a: a(x^2 + (b/a)x). Then complete the square inside. Example: Complete the square for x^2 - 10x. Half of -10 is -5. (-5)^2 = 25. x^2 - 10x + 25 = (x - 5)^2.

4. Parabola: Definition

A parabola is the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix). The vertex is midway between the focus and directrix. The axis of symmetry passes through the focus and vertex, perpendicular to the directrix. For a parabola with vertex at the origin (0, 0) and focus at (0, p): - Opens UP if p > 0, DOWN if p < 0. - Equation: x^2 = 4py. - Directrix: y = -p. For a horizontal parabola with focus at (p, 0): - Opens RIGHT if p > 0, LEFT if p < 0. - Equation: y^2 = 4px. - Directrix: x = -p.

5. Parabola Examples: Vertex at Origin

Example 1: x^2 = 8y. 4p = 8, so p = 2. Vertex: (0,0). Focus: (0, 2). Directrix: y = -2. Opens up. Example 2: y^2 = -12x. 4p = -12, so p = -3. Vertex: (0,0). Focus: (-3, 0). Directrix: x = 3. Opens left. Example 3: Write the equation of a parabola with vertex (0,0) and focus (0, -5). p = -5. x^2 = 4(-5)y = -20y.

6. Vertex Form of a Parabola

When the vertex is at (h, k): - Vertical: y = a(x - h)^2 + k (or equivalently (x-h)^2 = 4p(y-k)). - Horizontal: x = a(y - k)^2 + h. For y = a(x - h)^2 + k: - Vertex at (h, k). - Opens up if a > 0, down if a < 0. - p = 1/(4a). Focus at (h, k + 1/(4a)). Directrix: y = k - 1/(4a). Example: f(x) = 2(x - 1)^2 - 3. Vertex (1, -3), a = 2, opens up. p = 1/8. Focus (1, -3 + 1/8) = (1, -23/8).

7. Converting to Vertex Form

Complete the square to convert f(x) = ax^2 + bx + c to vertex form. Example: Convert f(x) = 2x^2 - 8x + 5 to vertex form. Factor 2: f(x) = 2(x^2 - 4x) + 5. Complete the square: 2(x^2 - 4x + 4 - 4) + 5 = 2(x-2)^2 - 8 + 5 = 2(x-2)^2 - 3. Vertex: (2, -3). Opens up.

8. Graphing Circles

Steps to graph (x - h)^2 + (y - k)^2 = r^2: 1. Plot the center (h, k). 2. Plot four points at distance r in each cardinal direction: (h+r, k), (h-r, k), (h, k+r), (h, k-r). 3. Connect with a smooth circle. Example: Graph (x-1)^2 + (y+2)^2 = 9. Center: (1, -2). Radius: 3. Points: (4, -2), (-2, -2), (1, 1), (1, -5).

9. Graphing Parabolas

Steps to graph y = a(x-h)^2 + k: 1. Plot the vertex (h, k). 2. Determine direction (up/down based on sign of a). 3. Plot additional points symmetrically around the axis x = h. 4. Draw the axis of symmetry (dashed vertical line through vertex). 5. Mark the focus and directrix if required. Example: Graph y = (x-2)^2 - 1. Vertex: (2, -1). Opens up. At x=3: y=0. At x=4: y=3. Axis: x = 2. Symmetric point at (1, 0).

10. Applications of Parabolas

Reflective property: any ray parallel to the axis of a parabola reflects through the focus. Satellite dish: a circular dish shaped like a paraboloid focuses all incoming signals at the focus point where the receiver is placed. Headlights: a bulb at the focus of a parabolic reflector sends light in parallel beams. Example: A parabolic dish has equation x^2 = 4y (in feet). Where should the receiver be placed? Focus at (0, 1). Receiver goes 1 foot above the vertex.

11. Common Mistakes

- Forgetting to add the completing-the-square constant to BOTH SIDES. - Reading center as (h, k) when the equation has (x + h) instead of (x - h): the center is -h. - Confusing x^2 = 4py (vertical, focus on y-axis) with y^2 = 4px (horizontal, focus on x-axis). - Missing that the vertex of y = (x-h)^2 + k is (h, k), not (-h, k). - Forgetting that the radius in the standard form is r, not r^2.