12.4 Conic Sections and Parabolas

Conic Sections Hyperbola Parabola Ellipse Circle

General Form Equation of Conic Sections Where ‘A’, ‘B’, ‘C’, ‘D’, ‘E’, and ‘F’ are real numbers AND ‘A’, ‘B’, and ‘C’ are not all zero.

Parabola as a function: Parabola: A “U” shaped curve opening either up or down. Vertex: The highest or lowest point on the parabola. Axis of Symmetry: the vertical line that divides the parabola into mirror images and passes through the vertex. What you’ve learned so far are parabolas that have vertical axes of symmetry Now we will learn equations of parabolas that have axes of symetry that are lines in either horizontal or vertical direction  the equation is not necessarily a function.

What is the general form of an equation of a parabola with: Vertex at the origin and opening upward? x-axis

Geometry of a Parabola Axis Directrix Point on parabola Distance to the directrix Distance to the focus Focus Vertex Distance to the directrix Distance to the focus = Directrix

Geometry of a Parabola Directrix Focus Distance to the directrix the focus = Directrix

Axis (0, P) P x-axis -P Directrix: y = -P

Axis Distance from (x, y) to (x, -p) = distance from (x, y) to (0, p) down: p < 0 (0, p) up: p > 0 p x-axis -p (x, -p) Directrix: y = -P

Your turn: 1. p = ? 2. p = ?

Axis Focal width (4p): the length of the chord passing thru the vertex and perpendicular to the axis. Focal length (p): the distance from the vertex to the focus. Axis focus p Vertex x-axis Directrix: y = -P

What is the focal length of the following parabola? What is the focal width of the above parabola? Axis Focal width (4p): the length of the chord passing thru the vertex and perpendicular to the axis. Focal length (p): the distance from the vertex to the focus. focus p Vertex x-axis Directrix: y = -P

y-axis Parabolas opening to the left or right are inverses of the parabolas opening up or down. Exchange ‘x’ and ‘y’. y-axis (P, 0) x-axis -P P Directrix: x = -P

Your turn: 3. Which way does the parabola open? 6. What is the focal length of the following parabola? 7. What is the focal width of the following parabola?

Parabolas with Vertex (0, 0) Standard Equations: Opening Direction: Up/down Right/Left Focus: (0, p) (p, 0) Directrix: Axis: y-axis x-axis Focal Length: Focal width:

Example Problem: Standard Equations: Opening Direction: down Focus: (0, -1/8) Directrix: Axis: y-axis Focal Length: Focal width:

Your turn: for the following equation, find the following information about the parabola. 8. Opening Direction: 9. Focus: 10. Directrix: 12. Axis: 13. Focal Length: 14. Focal width:

What is the equation of the parabola? Given the parabola with the following information: Which way does this parabola open? To the right Focus: (3, 0) Focal Length = ? p = 3 (0, 0) Focal length (p): the distance from the vertex to the focus. Directrix: x = -3 What is the equation of the parabola? Focal width = ? Focal width = 12 Focal width (4p): the length of the chord passing thru the vertex and perpendicular to the axis. What is the parabola’s axis ? x -axis

15. Which way does this parabola open? Your turn: Given the following information on a parabola: Directrix: y = 2 15. Which way does this parabola open? down Focus: (0, -2) 16. Focal Length = ? p = 2 (0, 0) 17. Focal width = ? Focal width = 8 18. What is the parabola’s axis ? y -axis 19. What is the equation of the parabola?

Your turn: 20. Find the equation of the parabola that has a directrix of y = -4 and a focus of (0, 4).

Parent Function and transformations How is the second parabola a transformation of the parent function? Right 3, down 5 How is the second parabola a transformation of the parent function? Left 2, down 7

Parabolas with Vertex (h, k) Standard Equation: Opening Direction: Up/down Up/down Focus: (0, p) (h, k + p) Directrix: y-axis (x = 0) Axis: x = h Focal Length: Focal width:

Vertex: (h, k) y value of vertex and y value of focus are the same  axis is a horizontal line. Standard Equations: (5, 4) Opening Direction: Left/right right Focus: (h + p, k) (3 + p, 4) = (5, 4) (3, 4) p = 2 Directrix: x = 3 - 2 x = 1 Axis: y = k y = 4 Focal Length: 2 8 Focal width:

Your turn: 21. Find the equation of the parabola that has a vertex of (2, 5) and a focus of (2, -2). Vertex: (h, k) h = 2 k = 5 Focus below vertex: opens down Focus: (h, k + p) = (2, -2)  5 + p = -2 p = -7