mathpower tm 12, western edition 3.6.1 3.6 chapter 3 conics
TRANSCRIPT
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MATHPOWERTM 12, WESTERN EDITION 3.6.1
3.6Chapter 3 Conics
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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 ofsymmetry, which intersectsthe 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 |.
3.6.2
The Parabola
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The Standard Form of the Equation of a Parabola with Vertex (0, 0)
The equation of a parabola withvertex (0, 0) and focus on the x-axisis y2 = 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. 3.6.3
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The equation of a parabola withvertex (0, 0) and focus on the y-axisis x2 = 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.
3.6.4
The Standard Form of the Equation of a Parabola with Vertex (0, 0)
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A parabola has the equation y2 = -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 y2 = 4px.Hence, 4p = -8 p = -2.
The coordinates of thefocus 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
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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 y2 = 4px.p is equal to the distance from the vertex to the focus, therefore p = -6.
The equation of the parabola is y2 = -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 y2 = -20x.
3.6.6
Finding the Equation of a Parabola with Vertex (0, 0)
Since the focus is on the x-axis, the equation of the parabola is y2 = 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 x2 = 4py.p is equal to the distance from the vertex to the focus, therefore p = 3.
The equation of the parabola is x2 = 12y.
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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 Ax2 + Cy2 + Dx + Ey + F = 0where A = 0 or C = 0.
3.6.7
The Standard Form of the Equation with Vertex (h, k)
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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.
3.6.8
The Standard Form of the Equation with Vertex (h, k)
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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
y2 - 10y + 25 = -12x + 72y2 + 12x - 10y - 47 = 0 General form
(y - 5)2 = 4(-3)(x - 6)(y - 5)2 = -12(x - 6)
3.6.9
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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 equationand 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
x2 + 4x + 4 = 8y - 48x2 + 4x + 8y + 52 = 0 General form
3.6.10
Finding the Equations of Parabolas
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Find the coordinates of the vertex and focus, the equation of the directrix, the axis of symmetry, and the direction of opening of y2 - 8x - 2y - 15 = 0.
y2 - 8x - 2y - 15 = 0 y2 - 2y + _____ = 8x + 15 + _____1 1
(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
Standardform
3.6.11
Analyzing a Parabola
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Graphing a Parabola
y2 - 10x + 6y - 11 = 0
9 9y2 + 6y + _____ = 10x + 11 + _____
(y + 3)2 = 10x + 20(y + 3)2 = 10(x + 2)
y 3 10(x 2)
y 10(x 2) 3
3.6.12
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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., x2 + 5x + 6 = 0 x2 + 5x + 6 = 0 (x + 3)(x + 2) = 0x + 3 = 0 or x + 2 = 0 x = -3 x = -2The result is two vertical,parallel lines. 3.6.13
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3.6.14