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Transcript of 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
Warm UpWarm Up
Lesson PresentationLesson Presentation
Lesson QuizLesson Quiz
Holt Algebra 2
10-6 Identifying Conic Sections
Warm UpSolve by completing the square.
1. x2 + 6x = 91
2. 2x2 + 8x – 90 = 0
Holt Algebra 2
10-6 Identifying Conic Sections
Identify and transform conic functions.
Use the method of completing the square to identify and graph conic sections.
Objectives
Holt Algebra 2
10-6 Identifying Conic Sections
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.
Holt Algebra 2
10-6 Identifying Conic Sections
Holt Algebra 2
10-6 Identifying Conic Sections
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.
Holt Algebra 2
10-6 Identifying Conic Sections
Identify the conic section that each equation represents.
Example 1: Identifying Conic Sections in Standard Form
This equation is of the same form as a circle.
C.
Holt Algebra 2
10-6 Identifying Conic Sections
Identify the conic section that each equation represents.
Check It Out! Example 1
a. x2 + (y + 14)2 = 112
– = 1 (y – 6)2
22
(x – 1)2
212b.
Holt Algebra 2
10-6 Identifying Conic Sections
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.
Holt Algebra 2
10-6 Identifying Conic Sections
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.
Holt Algebra 2
10-6 Identifying Conic Sections
Identify the conic section that the equation represents.
Example 2B: Identifying Conic Sections in General Form
0
Identify the values for A, B, and C.
9x2 – 12xy + 4y2 + 6x – 8y = 0.
A = 9, B = –12, C = 4
B2 – 4AC
Substitute into B2 – 4AC.(–12)2 – 4(9)(4)
Simplify.
Because B2 – 4AC = 0, the equation represents a parabola.
Holt Algebra 2
10-6 Identifying Conic Sections
Identify the conic section that the equation represents.
Example 2C: Identifying Conic Sections in General Form
33
Identify the values for A, B, and C.
8x2 – 15xy + 6y2 + x – 8y + 12 = 0
A = 8, B = –15, C = 6
B2 – 4AC
Substitute into B2 – 4AC.(–15)2 – 4(8)(6)
Simplify.
Because B2 – 4AC > 0, the equation represents a hyperbola.
Holt Algebra 2
10-6 Identifying Conic Sections
Identify the conic section that the equation represents.
9x2 + 9y2 – 18x – 12y – 50 = 0
Check It Out! Example 2a
Holt Algebra 2
10-6 Identifying Conic Sections
Identify the conic section that the equation represents.
12x2 + 24xy + 12y2 + 25y = 0
Check It Out! Example 2b
Holt Algebra 2
10-6 Identifying Conic Sections
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.
Holt Algebra 2
10-6 Identifying Conic Sections
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
Holt Algebra 2
10-6 Identifying Conic Sections
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.
Holt Algebra 2
10-6 Identifying Conic Sections
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 + +
Find the standard form of the equation by completing the square. Then identify and graph each conic.
Holt Algebra 2
10-6 Identifying Conic Sections
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
Holt Algebra 2
10-6 Identifying Conic Sections
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
Holt Algebra 2
10-6 Identifying Conic Sections
Find the standard form of the equation by completing the square. Then identify and graph each conic.
y2 – 9x + 16y + 64 = 0
Check It Out! Example 3a
Holt Algebra 2
10-6 Identifying Conic Sections
Check It Out! Example 3a Continued
Holt Algebra 2
10-6 Identifying Conic Sections
16x2 + 9y2 – 128x + 108y + 436 = 0
Check It Out! Example 3b
Find the standard form of the equation by completing the square. Then identify and graph each conic.
Holt Algebra 2
10-6 Identifying Conic Sections
Check It Out! Example 3b Continued