Unit 5: Quadratic Functions

Transcription

1 Unit 5: Quadratic Functions LESSON #5: THE PARABOLA GEOMETRIC DEFINITION DIRECTRIX FOCUS LATUS RECTUM Geometric Definition of a Parabola Quadratic Functions Geometrically, a parabola is the set of all points in a plane that are equidistant from a fixed line, called the directrix, and a fixed point, called the focus, that is not on the line. Parabola 1

2 Geometric Definition of a Parabola Notice: The vertex is halfway between the focus and the directrix. The line that passes through the focus and is perpendicular to the directrix is the axis of symmetry of the parabola. The point of intersection of the parabola with its axis of symmetry is called the vertex of the parabola. How can we transform the geometric definition to a standard form? Given: Directrix: x p Focus: p, 0 Point on parabola: P x, y Point on directrix: M p, y Recall: The distance from P x, y to the directrix is equal to the distance to the focus p, 0. What do we know? d 1 d 2 Recall: Distance Formula d x 1 x 2 2 y 1 y 2 2 Parabola 2

3 How can we transform the geometric definition to a standard form? Find d 1 : Point on parabola: P x, y Point on directrix: M p, y Find d 2 : Point on parabola: P x, y Focus: p, 0 Then, set d 1 d 2, and simplify. d 1 x p 2 y y 2 x p 2 d 2 x p 2 y 0 2 x p 2 y 2 d 1 d 2 : x p 2 x p 2 y 2 x p 2 x p 2 y 2 x 2 2xp p 2 x 2 2xp p 2 y 2 Equation of the Parabola: 4xp y 2 Standard Forms of the Equations of a Parabola: Vertex at the Origin Parabola with x axis symmetry Vertex Directrix Focus Opens Right Opens Left y 2 4px 0, 0 x p p, 0 p 0 p 0 Parabola 3

4 Standard Forms of the Equations of a Parabola: Vertex at the Origin Parabola with y axis symmetry Vertex Directrix Focus Opens Upward Opens Downward 0, 0 y p 0, p p 0 p 0 Latus Rectum of a Parabola The latus rectum of a parabola is a line segment that passes through its focus, is parallel to its directrix, and has its endpoints on the parabola. [Needed to determine width of parabola or how it opens] Length of latus rectum is 4p for y 4pxand x 4py. Parabola 4

5 Ex 1) Determine in which direction the parabola opens. Find the focus and directrix of the parabola given below. Sketch the parabola. V F y 2 12x y 2 4px Opens right since p is positive & y 2 Vertex: 0, 0 Need p value: 4p 12 p 3 Focus: 3, 0 Focus: 3 right of Vertex Directrix: Directrix: x 3 3 left of Vertex Use the length of the latus rectum to find two additional points on the parabola: latus rectum 4p One point 6 units above focus One point 6 units below focus Ex 2) Determine in which direction the parabola opens. Find the focus and directrix of the parabola given below. Sketch the parabola. V F x 2 8y Opens down since p is negative & x 2 Vertex: 0, 0 Need p value: 4p 8 p 2 Focus: 0, 2 Focus: 2 below Vertex Directrix: Directrix: y 2 2 above Vertex Latus rectum: 4p One point 4 units left of focus One point 4 units right of focus Parabola 5

6 Recall: Vertex Form of a Parabola Recall from Lesson #4: Vertex Form of a Parabola f x a x h k Vertex at h, k Parent Function The graph at the right completed a series of transformations of the parent function Parabola with vertical symmetry Vertex Directrix Focus Standard Forms of the Equations of a Parabola: Vertex at h, k Recall: Parent Function Horizontal translations Replace x with x h Vertical translations Replace y with y k x h 2 4p y k Note: BOTH h and k are opposite of what they appear! x h 2 4p y k h, k y k p h, k p Parabola 6

7 Parabola with horizontal symmetry Vertex Directrix Focus Standard Forms of the Equations of a Parabola: Vertex at h, k Recall: Parent Function y 2 4px Horizontal translations Replace x with x h Vertical translations Replace y with y k y k 2 4p x h Note: BOTH h and k are opposite of what they appear! y k 2 4p x h h, k x h p h p,k Ex 1) Determine in which direction the parabola opens. Find the vertex, focus, and directrix of the parabola given below. Sketch the parabola. V F x h 2 4p y k x y 1 Opens up since p is positive & x 2 Vertex: 3, 1 Need p value: 4p 8 p 2 Focus: 3, 1 Directrix: y 3 Latus rectum: 4p 2 units above vertex, since it opens up. Horizontal line 2 units below vertex, since it opens up One point 4 units left of focus One point 4 units right of focus Parabola 7

8 Ex 2) Determine in which direction the parabola opens. Find the vertex, focus, and directrix of the parabola given below. Sketch the parabola. F V y k 2 4p x h y x 2 Opens left since p is negative & y 2 Vertex: 2, 1 Need p value: 4p 12 p 3 Focus: 1, 1 Directrix: x 5 Latus rectum: 3 units left of vertex, since it opens left. Vertical line 3 units right of vertex, since it opens left One point 6 units above of focus One point 6 units below of focus Ex 3) Find the vertex, focus, and directrix of the parabola given below. x 2 2x 4y 9 0 Goal: x h 2 4p y k x 2 2x 1 4y 9 1 x 1 2 4y 8 x y 2 Opens up since p is positive & x 2 Need p value: 4p 4 p 1 Vertex: 1, 2 Focus: 1, 3 Directrix: y 1 How? Complete the Square! 1 unit above vertex, since it opens up. Horizontal line 1 unit below vertex, since it opens up. Parabola 8

9 Ex 4) Find the vertex, focus, and directrix of the parabola given below. Goal: y k 2 4p x h y 2 2y 12x 23 0 y 2 2y 1 12x 23 1 y x 24 y x 2 Complete the Square! Opens left since p is negative & y 2 Need p value: 4p 12 p 3 Vertex: 2, 1 Focus: 1, 1 Directrix: x 5 3 units left of vertex, since it opens left. Vertical line 3 units right of vertex, since it opens left. Writing Equations of Parabolas Given Conditions Recall: Parabola with Vertex at the origin y 2 4px Need the vertex and the value of p to write an equation! Parabola 9

10 Ex 1) Write the standard form of the equation of a parabola given focus 8,0 and directrix x 8. Total distance between directrix and focus is 16 units. Recall: vertex is equidistant between the directrix and focus. Vertex is 8 units from directrix and 8 units focus. Vertex: 0, 0 Since focus is to the right of the vertex, parabola opens right. p is positive & y 2 p 8 y 2 4px y 2 4 8x y 2 32x Ex 2) Write the standard form of the equation of a parabola given focus 0, 15 and directrix y 15. Total distance between directrix and focus is 30 units. Vertex is 15 units from directrix and 15 units focus. Vertex: 0, 0 Since focus is below the vertex, parabola opens down. p is negative & x 2 p 15 x y x 2 60y Parabola 10

11 Ex 3) Write the standard form of the equation of a parabola given vertex 5, 2 and focus 7, 2. Total distance between vertex and focus is 2 units. Since focus is to the right the vertex, parabola opens right. p is positive & y 2 p 2 y k 2 4p x h y y x 5 x 5 Ex 4) Write the standard form of the equation of a parabola given vertex 2, 3 and focus 2, 5. Total distance between vertex and focus is 2 units. Since focus is to below the vertex, parabola opens down. p is negative & x 2 p 2 x h 2 4p y k x y 3 x y 3 Parabola 11

12 Ex 5) Write the standard form of the equation of a parabola given focus 2,4 and directrix x 4. Total distance between directrix and focus is 6 units. Vertex is 3 units from directrix and 3 units focus. Vertex: 1, 4 Since focus is to the right the vertex, parabola opens right. p is positive & y 2 p 3 y k 2 4p x h y x 1 y x 1 Ex 6) Write the standard form of the equation of a parabola given focus 7, 1 and directrix y 9. Total distance between directrix and focus is 8 units. Vertex is 4 units from directrix and 3 units focus. Vertex: 7, 5 Since focus is above the vertex, parabola opens up. p is positive & x 2 p 4 x h 2 4p y k x y 5 x y 5 Parabola 12