Lecture 2b Parabolas.pdf
- 1. WHAT DO YOU THINK???
- 2. Analytic Geometry Lecture 2: Parabolas Noven S. Villaber Fatima College of Camiguin, Inc. SHS Department PRECALCUS 16
- 3. Learning Outcomes of the Lesson At the end of the lesson, the student is able to: (1) define a parabola; (2) determine the standard form of equation of a parabola; (3) graph a parabola in a rectangular coordinate system
- 4. Parabolas The parabola is the locus of all points in a plane that are the same distance from a line in the plane, the directrix, as from a fixed point in the plane, the focus. | p | | p | The parabola has one axis of symmetry, which intersects the parabola at its vertex. The distance from the vertex to the focus is | p |. The distance from the directrix to the vertex is also | p |. Point Focus = Point Directrix PF = PD
- 6. Parts of Parabola Fixed line = Directrix Fixed point = Focus The line that passes through the focus and is perpendicular to the directrix = axis of symmetry The point where the parabola intersects with its axis of symmetry (it is point midway between the latus rectum and the directrix = vertex Line segment that passes through the focus of a parabola and is perpendicular to the axis of symmetry = Latus Rectum/focal width Parabola has 4 openings (upward, downward, to the right, and to the left).
- 8. The general form of the equation is where D, E, and F are constants
- 9. Examples 23
- 10. The Standard Form of the Equation with Vertex (h, k) 24 Parabola Axis of Symmetry Opening Focus Latus Rectum Directrix Horizontal Right, if p > 0 Left, if p<0 (h+p,k) Length: 4p Equation: =ℎ+ Endpoints: (ℎ+ , ±2 ) Equation: =ℎ− Vertical Upward, if p > 0 Downward , if p<0 (h,k+p) Length: 4p Equation: = + Endpoints: (ℎ±2 , + ) Equation: = −
- 11. General Form To Standard Form Steps: 1.Group the terms with same variables. 2.Move the remaining terms to the right side of the equation. 3.Create a perfect square trinomial by completing the squares. 4.Simplify both sides of the equation. 5.Express the perfect square trinomials as square of binomials to make the equation in the vertex form.
- 12. Example 26
- 14. Standard Form To General Form
- 15. Example 29
- 17. Standard form of a Parabola with vertex at the origin (0,0) 31 Parabola Axis of Symmetry Opening Focus Latus Rectum Directrix Horizontal Right, if p > 0 Left, if p<0 (p,0) Length: 4p Equation: =± Endpoints: ( ,±2 ) Equation: =− Vertical Upward, if p > 0 Downward, if p<0 (0,p) Length: 4p Equation: =± Endpoints: ( ,±2 ) Equation: =−
- 18. Example
- 29. Graph of Parabola To graph at Vertex (h,k): 1. Locate the vertex of the parabola. 2. Determine its axis of symmetry by considering the variable with the second degree 3. Identify the opening of the parabola by examining the value of its focus (p). 4. Locate the focus of the parabola. 5. Locate the endpoints of the latus rectum. 6. Plot the directrix.
- 30. Example
- 32. Step 4 Locate the focus of the parabola. Since p = 1, the focus of the parabola is 1 unit to the right of the vertex. Thus, the focus is at (h+p,k) = (2+1, -3) = (3, -3) Step 5 Locate the endpoints of the latus rectum. To locate the endpoints of the latus rectum ± ± This means that from the focus (3,-3), we move 2 units upward and another 2 units downward. Therefore, the endpoints of the latus rectum are located at (3,-1) and (3,-5). Also, the length of the latus rectum is 4 units.
- 33. Step 6 Plot the directrix.
- 34. Graph of Parabola To graph at Vertex (0,0): 1. Locate the vertex of the parabola. 2. Determine its axis of symmetry by considering the variable with the second degree 3. Identify the opening of the parabola by examining the value of its focus (p). 4. Locate the focus of the parabola. 5. Locate the endpoints of the latus rectum. 6. Plot the directrix.
- 35. Example
- 36. Step 1 Step 2 Step 3 Step 4 Locate the focus of the parabola. Since p = 5, the focus of the parabola is 5 units above the vertex. Thus, the focus is at (0,5) 50
- 37. 51 Step 5 Locate the endpoints of the latus rectum. To locate the endpoints of the latus rectum ±2 =±10. This means that from the focus (0,5), we move 10 units to the right and another 10 units to the left. Therefore, the endpoints of the latus rectum are located at (-10,5) and (10,5). Also, the length of the latus rectum is 20 units, which is actually the value of 4p (the coefficients of y in the given example). Step 6 Plot the directrix.