Warm Up Define a parabola Define vertex Define

Warm Up Define a parabola. Define vertex. Define directrix.

Parabolas A parabola is a set of points in a plane that are equidistant from a fixed line, the directrix, and a fixed point, the focus. Q Parabolas make a U shape. d 1 Directrix For any point Q that is on the parabola, d 2 = d 1 d 2 Focus

Things you should know about a parabola. Forms of equations: y = a(x – h)2 + k opens up if a is positive opens down if a is negative vertex is (h, k) V y = ax 2 + bx + c opens up if a is positive Axis of symmetry cuts the parabola into 2 equal pieces – goes through the focus opens down if a is negative vertex is

Dir rix ect Focus V Horizontal Parabola – Opens right or left Axis of symmetry is horizontal y-variable is squared Axis of Symmetry x = a(y – k)2 + h or x = ay 2 + by + c

Vertical Parabola Horizontal Parabola Axis of symmetry y= # (y – k)2 = 4 p(x – h) Standard Form Vertex: (h, k) x= # (x – h)2 = 4 p(y – k) Vertex: (h, k) If a > 0 or 4 p > 0, opens right If a > 0 or 4 p > 0, opens up If a < 0 or 4 p < 0, opens left If a < 0 or 4 p < 0, opens down The directrix is vertical The directrix is horizontal |p| is the distance from the vertex to the focus The directrix is the same distance from the vertex as the focus.

Find the standard form of the equation of the parabola given: The focus is (2, 4) and the directrix is x = - 4 The vertex is between the focus and directrix vertex is (-1, 4) Directrix is Vertical = Parabola is Horizontal Focus on inside = V Opens to the right Equation: (y – k)2 = 4 p(x – h) |p| = 3 Equation: (y – 4)2 = 12(x + 1) F

Find the standard form of the equation of the parabola given: The focus is (-3, 3) and the directrix is y = 1 vertex is (-3, 2) Directrix is Horizontal = Parabola is Vertical Focus on inside = Opens up Equation (x – h)2 = 4 p(y – k) |p| = 1 Equation: (x + 3)2 = 4(y – 2) F V

Find the standard form of the equation of the parabola and the directrix given: the vertex is (2, -3) and focus is (2, -5) The location of the vertex and focus means – it must be a vertical parabola that opens down Equation: (x – h)2 = 4 p(y – k) V F |p| = 2 Equation: (x – 2)2 = -8(y + 3) The vertex is equidistant from the focus and directrix, so the directrix is y = -1

A satellite dish is in the shape of a parabolic surface. The dish is 12 ft in diameter and 2 ft deep. How far from the base should the receiver be placed? 12 2 (Looking for the focus…) (-6, 2) Since the parabola is vertical it’s equation must be of the form: (x – h)2 = 4 p(y – k) Vertex (0, 0) x 2 = 4 py Plug in (6, 2) 36 = 4 p(2) 36 = 8 p p = 4. 5 The receiver should be placed 4. 5 feet above the base of the dish. (6, 2)

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Parabolas! (y + 3)2 = 4(x + 1) Find the vertex, focus, axis of symmetry, and directrix. Then graph the parabola Vertex: (-1, -3) Ao. S: y = -3 4 p = 4 p=1 The parabola is horizontal and opens to the right Focus: (0, -3) Directrix: x = -2 V F x 3 0 0 y -7 -5 -1 3 1 Pick some numbers around the vertex.

Parabolas! (x + 2)2 = 8(y - 1) Find the vertex, the axis of symmetry, focus and directrix. Then graph the parabola Vertex: (-2, 1) Ao. S: x = -2 4 p = 8 p=2 F V Focus: (-2, 3) Directrix: y = -1 x y -4 -3 -1 0 1. 5 1. 1 1. 5 The parabola is vertical and opens up

Find the vertex, the axis of symmetry, the focus, the directrix, and graph. (x – 1)2 = -(y – 4) (x – 1)2 = -1(y – 4) Vertex: (1, 4) Ao. S: x = 1 V F The parabola is vertical and opens down 4 p = 1 p =. 25 Focus: (1, 3. 75) Directrix: y = 4. 25 x y -1 0 0 3 2 3 3 0

(400, 160) (300, h) 300 100 The towers of a suspension bridge are 800 ft apart and rise 160 ft above the road. The cable between them has the shape of a parabola, and the cable just touches the road midway between the towers. What is the height of the cable 100 ft from a tower? Since the parabola is vertical and has its vertex at (0, 0) its equation must be of the form: (x – h)2 = 4 p(y – k) x 2 = 4 py At (400, 160), 160, 000 = 4 p(160) 1000 = 4 p p = 250 thus the equation is x 2 = 1000 y At (300, h), 90, 000 = 1000 h h = 90 The cable would be 90 ft high at a point 100 ft from a tower.