Copyright © Cengage Learning. All rights reserved.

Pre-Calculus Honors9.1: Circles and Parabolas

HW: p.643 (8-24 even, 28, 30, 36, 42)

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Conics

A conic section (or simply conic)is the intersection of a plane anda double-napped cone.

Notice in Figure 9.1 that in theformation of the four basic conics,the intersecting plane does notpass through the vertex of the cone.

Figure 9.1

Basic Conics

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ConicsThe definition of a circle is the collection of all points (x, y) that are equidistant from a fixed point (h, k) leads to the standard equation of a circle

(x – h)2 + (y – k)2 = r2. Equation of circle

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Example 1 – Finding the Standard Equation of a Circle

The point (1, 4) is on a circle whose center is at (–2, –3), as shown in Figure 9.4. Write the standard form of the equation of the circle.

Figure 9.4

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Example 2 – Sketching a CircleSketch the circle given by the equation

x2 – 6x + y2 – 2y + 6 = 0and identify its center and radius.Solution:Begin by writing the equation in standard form.

x2 – 6x + y2 – 2y + 6 = 0

(x2 – 6x + __) + (y2 – 2y + __) = –6 + __ + __ Complete the squares

In this form, you can identify the center and radius of the circle and then sketch the graph.

Write in standard form

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Example 2 – SolutionNow graph the function.

cont’d

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Practice with circles.1. Write the equation of a circle in standard form with

center = (3, 7) and point on the circle = (1, 0).2. Write the equation of a circle in standard form with

center = (-3, -1) and diameter = Identify the center and radius of the circle, then graph. 3. (x – 3)2 + y2 = 84. x2 – 14x + y2 + 8y + 40 = 0

cont’d

72

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Do NowGraph 9x2 + 9y2 + 54x – 36y + 17 = 0

9Copyright © Cengage Learning. All rights reserved.

Pre-Calculus Honors9.1: Circles and Parabolas

HW: p.644 (70 and 76: write in standard form), (62, 66, 68, 78: graph labeling the vertex and 2

additional points, determine domain and range).

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Parabolas

Vertex: (h, k); there are other characteristics of a parabola we are not going to find and sketch in the graph (directrix and focus).

opens up or down b/c x is squared

opens right or left b/c y is squared

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ParabolasRewrite the parabola in standard form:

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ParabolasSolution: Convert to standard form by completing the square.

–2y = x2 + 2x – 1

1 – 2y = x2 + 2x

Create a coefficient of 1 for x2

Isolate the x2 and bx term

__ + 1 – 2y = x2 + 2x + __

2 – 2y = x2 + 2x + 1

–2(y – 1) = (x + 1)2

Complete the square

Combine like terms

Factor to write in standard form

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Graph the parabola.Graph the parabola. Label the vertex and 2 additional points on the parabola. Determine the domain and range.1. y2 = 3x

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Graph the parabola.Graph the parabola. Label the vertex and 2 additional points on the parabola. Determine the domain and range.2. (x + ½)2 = 4(y – 1)

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Graph the parabola.Graph the parabola. Label the vertex and 2 additional points on the parabola. Determine the domain and range.3. y2 + x + y = 0

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Graph the parabola.Graph the parabola. Label the vertex and 2 additional points on the parabola. Determine the domain and range.4. 3x2 + 6x + y = 4