holt algebra 2 10-6 identifying conic sections 10-6 identifying conic sections holt algebra 2 warm...

Holt Algebra 2

10-6 Identifying Conic Sections10-6 Identifying Conic Sections

Holt Algebra 2

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Holt Algebra 2

10-6 Identifying Conic Sections

Warm UpSolve by completing the square.

1. x2 + 6x = 91

2. 2x2 + 8x – 90 = 0

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Identify and transform conic functions.

Use the method of completing the square to identify and graph conic sections.

Objectives

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In Lesson 10-2 through 10-5, you learned about the four conic sections. Recall the equations of conic sections in standard form. In these forms, the characteristics of the conic sections can be identified.

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Holt Algebra 2


Identify the conic section that each equation represents.

Example 1: Identifying Conic Sections in Standard Form

A.

This equation is of the same form as a parabola with a horizontal axis of symmetry.

x + 4 = (y – 2)2

10

B.

This equation is of the same form as a hyperbola with a horizontal transverse axis.

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Example 1: Identifying Conic Sections in Standard Form

This equation is of the same form as a circle.

C.

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Check It Out! Example 1

a. x2 + (y + 14)2 = 112

– = 1 (y – 6)2

22

(x – 1)2

212b.

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All conic sections can be written in the general form Ax2 + Bxy + Cy2 + Dx + Ey+ F = 0. The conic section represented by an equation in general form can be determined by the coefficients.

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Identify the conic section that the equation represents.

Example 2A: Identifying Conic Sections in General Form

20

Identify the values for A, B, and C.

4x2 – 10xy + 5y2 + 12x + 20y = 0

A = 4, B = –10, C = 5

B2 – 4AC

Substitute into B2 – 4AC.(–10)2 – 4(4)(5)

Simplify.

Because B2 – 4AC > 0, the equation represents a hyperbola.

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Example 2B: Identifying Conic Sections in General Form

0


9x2 – 12xy + 4y2 + 6x – 8y = 0.

A = 9, B = –12, C = 4

B2 – 4AC


Simplify.

Because B2 – 4AC = 0, the equation represents a parabola.

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Example 2C: Identifying Conic Sections in General Form

33


8x2 – 15xy + 6y2 + x – 8y + 12 = 0

A = 8, B = –15, C = 6

B2 – 4AC


Simplify.

Because B2 – 4AC > 0, the equation represents a hyperbola.

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9x2 + 9y2 – 18x – 12y – 50 = 0

Check It Out! Example 2a

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12x2 + 24xy + 12y2 + 25y = 0

Check It Out! Example 2b

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You must factor out the leading coefficient of x2 and y2 before completing the square.

Remember!

If you are given the equation of a conic in standard form, you can write the equation in general form by expanding the binomials.

If you are given the general form of a conic section, you can use the method of completing the square from Lesson 5-4 to write the equation in standard form.

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Find the standard form of the equation by completing the square. Then identify and graph each conic.

Example 3A: Finding the Standard Form of the Equation for a Conic Section

Rearrange to prepare for completing the square in x and y.

x2 + y2 + 8x – 10y – 8 = 0

x2 + 8x + + y2 – 10y + = 8 + +

Complete both squares.2

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Example 3A Continued

(x + 4)2 + (y – 5)2 = 49 Factor and simplify.

Because the conic is of the form (x – h)2 + (y – k)2 = r2, it is a circle with center (–4, 5) and radius 7.

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Example 3B: Finding the Standard Form of the Equation for a Conic Section

Rearrange to prepare for completing the square in x and y.

5x2 + 20y2 + 30x + 40y – 15 = 0

5x2 + 30x + + 20y2 + 40y + = 15 + +

Factor 5 from the x terms, and factor 20 from the y terms.

5(x2 + 6x + )+ 20(y2 + 2y + ) = 15 + +


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Example 3B Continued

Complete both squares.

5(x + 3)2 + 20(y + 1)2 = 80 Factor and simplify.

Divide both sides by 80.

65 x2 + 6x + + 20 y2 + 2y + = 15 + 5 + 20

2

2

2

2

6

2

2

2

2 2

1

16 4

x + 3 2 y +1 2

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Because the conic is of the form (x – h)2

a2+ = 1,(y – k)2

b2

it is an ellipse with center (–3, –1), horizontal major axis length 8, and minor axis length 4. The co-vertices are (–3, –3) and (–3, 1), and the vertices are (–7, –1) and (1, –1).

Example 3B Continued

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y2 – 9x + 16y + 64 = 0

Check It Out! Example 3a

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Check It Out! Example 3a Continued

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16x2 + 9y2 – 128x + 108y + 436 = 0

Check It Out! Example 3b


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Check It Out! Example 3b Continued

holt algebra 2 10-6 identifying conic sections 10-6 identifying conic sections holt algebra 2 warm...

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