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MATHPOWER TM 12, WESTERN EDITION 3.6.1 3.6 Chapter 3 Conics.

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Presentation on theme: "MATHPOWER TM 12, WESTERN EDITION 3.6.1 3.6 Chapter 3 Conics."— Presentation transcript:

1 MATHPOWER TM 12, WESTERN EDITION Chapter 3 Conics

2 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. Point Focus = Point Directrix PF = PD The parabola has one axis of symmetry, which intersects the parabola at its vertex. | p | The distance from the vertex to the focus is | p |. The distance from the directrix to the vertex is also | p | The Parabola

3 The Standard Form of the Equation of a Parabola with Vertex (0, 0) The equation of a parabola with vertex (0, 0) and focus on the x-axis is y 2 = 4px. The coordinates of the focus are (p, 0). The equation of the directrix is x = -p. If p > 0, the parabola opens right. If p < 0, the parabola opens left

4 The equation of a parabola with vertex (0, 0) and focus on the y-axis is x 2 = 4py. The coordinates of the focus are (0, p). The equation of the directrix is y = -p. If p > 0, the parabola opens up. If p < 0, the parabola opens down The Standard Form of the Equation of a Parabola with Vertex (0, 0)

5 A parabola has the equation y 2 = -8x. Sketch the parabola showing the coordinates of the focus and the equation of the directrix. The vertex of the parabola is (0, 0). The focus is on the x-axis. Therefore, the standard equation is y 2 = 4px. Hence, 4p = -8 p = -2. The coordinates of the focus are (-2, 0). The equation of the directrix is x = -p, therefore, x = 2. F(-2, 0) x = 2 Sketching a Parabola 3.6.5

6 A parabola has vertex (0, 0) and the focus on an axis. Write the equation of each parabola. Since the focus is (-6, 0), the equation of the parabola is y 2 = 4px. p is equal to the distance from the vertex to the focus, therefore p = -6. The equation of the parabola is y 2 = -24x. b) The directrix is defined by x = 5. The equation of the directrix is x = -p, therefore -p = 5 or p = -5. The equation of the parabola is y 2 = -20x Finding the Equation of a Parabola with Vertex (0, 0) Since the focus is on the x-axis, the equation of the parabola is y 2 = 4px. c) The focus is (0, 3). a) The focus is (-6, 0). Since the focus is (0, 3), the equation of the parabola is x 2 = 4py. p is equal to the distance from the vertex to the focus, therefore p = 3. The equation of the parabola is x 2 = 12y.

7 For a parabola with the axis of symmetry parallel to the y-axis and vertex at (h, k): The equation of the axis of symmetry is x = h. The coordinates of the focus are (h, k + p). The equation of the directrix is y = k - p. When p is positive, the parabola opens upward. When p is negative, the parabola opens downward. The standard form for parabolas parallel to the y-axis is: (x - h) 2 = 4p(y - k) The general form of the parabola is Ax 2 + Cy 2 + Dx + Ey + F = 0 where A = 0 or C = The Standard Form of the Equation with Vertex (h, k)

8 For a parabola with an axis of symmetry parallel to the x-axis and a vertex at (h, k): The equation of the axis of symmetry is y = k. The coordinates of the focus are (h + p, k). The equation of the directrix is x = h - p. The standard form for parabolas parallel to the x-axis is: (y - k) 2 = 4p(x - h) When p is negative, the parabola opens to the left. When p is positive, the parabola opens to the right The Standard Form of the Equation with Vertex (h, k)

9 Finding the Equations of Parabolas Write the equation of the parabola with a focus at (3, 5) and the directrix at x = 9, in standard form and general form The distance from the focus to the directrix is 6 units, therefore, 2p = -6, p = -3. Thus, the vertex is (6, 5). (6, 5) The axis of symmetry is parallel to the x-axis: (y - k) 2 = 4p(x - h)h = 6 and k = 5 Standard form y y + 25 = -12x + 72 y x - 10y - 47 = 0 General form (y - 5) 2 = 4(-3)(x - 6) (y - 5) 2 = -12(x - 6) 3.6.9

10 Find the equation of the parabola that has a minimum at (-2, 6) and passes through the point (2, 8). The axis of symmetry is parallel to the y-axis. The vertex is (-2, 6), therefore, h = -2 and k = 6. Substitute into the standard form of the equation and solve for p: (x - h) 2 = 4p(y - k) (2 - (-2)) 2 = 4p(8 - 6) 16 = 8p 2 = p x = 2 and y = 8 (x - h) 2 = 4p(y - k) (x - (-2)) 2 = 4(2)(y - 6) (x + 2) 2 = 8(y - 6) Standard form x 2 + 4x + 4 = 8y - 48 x 2 + 4x + 8y + 52 = 0 General form Finding the Equations of Parabolas

11 Find the coordinates of the vertex and focus, the equation of the directrix, the axis of symmetry, and the direction of opening of y 2 - 8x - 2y - 15 = 0. y 2 - 8x - 2y - 15 = 0 y 2 - 2y + _____ = 8x _____ 11 (y - 1) 2 = 8x + 16 (y - 1) 2 = 8(x + 2) The vertex is (-2, 1). The focus is (0, 1). The equation of the directrix is x + 4 = 0. The axis of symmetry is y - 1 = 0. The parabola opens to the right. 4p = 8 p = 2 Standard form Analyzing a Parabola

12 Graphing a Parabola y x + 6y - 11 = 0 99y 2 + 6y + _____ = 10x _____ (y + 3) 2 = 10x + 20 (y + 3) 2 = 10(x + 2)

13 General Effects of the Parameters A and C When A x C = 0, the resulting conic is an parabola. When A is zero: If C is positive, the parabola opens to the left. If C is negative, the parabola opens to the right. When A = D = 0, or when C = E = 0, a degenerate occurs. When C is zero: If A is positive, the parabola opens up. If A is negative, the parabola opens down. E.g., x 2 + 5x + 6 = 0 x 2 + 5x + 6 = 0 (x + 3)(x + 2) = 0 x + 3 = 0 or x + 2 = 0 x = -3 x = -2 The result is two vertical, parallel lines

14 3.6.14


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