Parabolas

Definitions Parabola – set of all points equidistant from a fixed line (directrix) and a fixed point (focus) Vertex – midpoint of segment from focus  to directrix. Axis of Symmetry: line through the focus and vertex Vertical Parabola Form: (x – h) 2 = 4p(y – k) Horizontal Parabola form: (y – k) 2 = 4p(x – h)

FOCUS FOCUS VERTEX VERTEX DIRECTRIX DIRECTRIX VERTICAL PARABOLA HORIZONTAL PARABOLA Opens either up or down. Up if p > 0, down if p 0, down if p < 0. Opens either left or right. Right if p > 0, left if p 0, left if p < 0.

p: distance from the vertex to the focus and from the vertex to the directrix p p 2p Important note: The points on a parabola are symmetric across the Axis of Symmetry. So, if you know one point, you can find the other. Focal Chord – the line segment that goes through the focus, and has endpoints on the parabola. It’s length is 4p.

Ex 1: Write the equation of the parabola in standard form.

Decide whether the parabola has vertical or horizontal axis of symmetry, and tell which way the graph opens x 2 = 3y 3. (y – 4) 2 = 3x y = (x + 1) 2 Vertical Opens down Horizontal Opens right Vertical Opens up HINT: Which variable is squared?

Given the following information, write the equation of the parabola 5.Vertex (-2, 5); p= -½; Vertical Axis of Symmetry 6.Vertex (1, -3); p = 1 / 8 ; Vertical Axis of Symmetry Write the formula, and fill in values

Given the following information, write the equation of the parabola 7. Vertex (6, -1); p = - 1 / 12 ; Horizontal Axis of Symmetry 8. Vertex (-5, -7); p = 1; Horizontal Axis of Symmetry Write the formula, and fill in values

Vertex: Focus: Directrix: Axis: Equation: (0,0) (0,3) y = -3 x = 0 What is p? So, 4p is…? Is this a horizontal or vertical parabola? Write the formula, and fill in values. 3 Vertical positive or negative? Why? 9

Vertex: Focus: Directrix: Axis: Equation: (0,0) (-2, 0) x = 2 y = 0 What is p? Is this a horizontal or vertical parabola? Write the formula, and fill in values. 2 Horizontal positive or negative? Why? 10

Vertex: Focus: Directrix: Axis: Equation: (3,2) (3,1) y = 3 x = 3 What is p? Is this a horizontal or vertical parabola? Write the formula, and fill in values. 1 Vertical positive or negative? Why? 11

Vertex: Focus: Directrix: Axis: Equation: (-4,2) (0,2) x = -8 y = 2 What is p? Is this a horizontal or vertical parabola? Write the formula, and fill in values. 4 Horizontal positive or negative? Why? 12

13. Find the Vertex, the Focus, the Directrix, and sketch the graph What is the vertex? What is p? Which way does the graph open? Where is the focus? Where is the directrix? Sketch the graph. (3, -1) 3 Up (3, 2) y = -4

14. Find the Vertex, the Focus, the Directrix, and sketch the graph What is the vertex? What is p? Which way does the graph open? Where is the focus? Where is the directrix? Sketch the graph. (3, 5) Left (2, 5) x = 4

15. Find the equation if… The Vertex is (-3, 6), and the Focus is (5, 6) What do we need for the equation? We need the vertex (GOT IT!) and p. Draw a sketch. How far away from the vertex is the focus? 8 Positive or Negative? Therefore, p = 8 and 4p = 32 So, the equation is:

16. Find the equation if… The Vertex is (2, -1), and the Directrix is x = 5 What do we need for the equation? We need the vertex (GOT IT!) and p. Draw a sketch. How far away from the vertex is the directrix? 3 Positive or Negative? Therefore, p = -3 and 4p = -12 So, the equation is:

17. Find the equation if… The Directrix is y = 5, and the Focus is (-3, 1) What do we need for the equation? We need the vertex and p. Draw a sketch. Where is the vertex in relation to the focus and directrix? Halfway in between Therefore, the vertex is at (-3, 3) Opens up or down? Therefore, p = -2 and 4p = -8 So, the equation is: Down

How does a Parabola Work? Anything entering the parabola is reflected to the focus, concentrating the signal.

Anything leaving from the focus reflects off the parabola in a straight line creating a beam. How does a Parabola Work?

18. Where are Parabolas used?

Where are Parabolas used? Cars - headlights Sports – the trajectory of a ball

Where are Parabolas used? Communication – a microphone Architecture – the St. Louis Arch