MATHPOWERTM 12, WESTERN EDITION 3.6.1

3.6Chapter 3 Conics

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

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

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)

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

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.

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)

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)

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

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

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

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

General Effects of the Parameters A and CWhen 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

Suggested Questions:Pages 167-169A 1, 3, 5, 7, 8 (general form)

3.6.14

B 9, 12, 13, 14, 16, 18, 27, 28, 29, 46, 47