Writing equations of conics in vertex form
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Transcript of Writing equations of conics in vertex form
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WRITING EQUATIONS OF CONICS IN VERTEX FORMMM3G2
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Write the equation for the circle in vertex form:
Example 1
Step 1: Move the constant to the other side of the equation & put your common variables together
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Example 1
Step 2: Identify the coefficients of the squared terms and divide everything by that coefficient. Both coefficients are 1 so divide
everything by 1
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Example 1
Step 3: Group the x terms together and the y terms together using parenthesis.
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Example 1
Step 4: Complete the square for the x terms
Then for the y terms
22=1 12=1 −42 =−2 (−2)2=4
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Example 1
Step 5: Write the factored form for the groups.
What is the center of this circle?
What is the radius?
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Write the equation for the circle in vertex form:
Example 2
Step 1: Move the constant to the other side of the equation & put your common variables together
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Example 2
Step 2: Identify the coefficients of the squared terms and divide everything by that coefficient. Both coefficients are 2 so divide
everything by 2
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Example 2
Step 3: Group the x terms together and the y terms together using parenthesis.
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Example 2
Step 4: Complete the square for the x terms
Then for the y terms
62=3 32=9 42=2 22=4
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Example 2
Step 5: Write the factored form for the groups.
What is the center of this circle?
What is the radius?
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Write the equation for the circle in vertex form:
Example 3
Step 1: Move the constant to the other side of the equation & put your common variables together
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Example 3
Step 2: Identify the coefficients of the squared terms and divide everything by that coefficient. Both coefficients are 4 so divide
everything by 4
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Example 3
Step 3: Group the x terms together and the y terms together using parenthesis.
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Example 3
Step 4: Complete the square for the x terms
Then for the y terms
62=3 32=9 82=4 42=16
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Example 3
Step 5: Write the factored form for the groups.
What is the center of this circle?
What is the radius?
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Write the equation for the circle in vertex form:
Example 4
Step 1: Move the constant to the other side of the equation & put your common variables together
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Example 4
Step 2: Identify the coefficients of the squared terms and divide everything by that coefficient. Both coefficients are 5 so divide
everything by 5
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Example 4
Step 3: Group the x terms together and the y terms together using parenthesis.
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Example 4
Step 4: Complete the square for the x terms
Then for the y terms
−162 =−8¿ 4
2=2 22=4
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Example 4
Step 5: Write the factored form for the groups.
What is the center of this circle?
What is the radius?
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Recall: The equation for a circle does not have
denominators The equation for an ellipse and a
hyperbola do have denominators The equation for a circle is not equal to
one The equation for an ellipse and a
hyperbola are equal to one We have a different set of steps for
converting ellipses and hyperbolas to the vertex form:
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Write the equation for the ellipse in vertex form:
Example 5
Step 1: Move the constant to the other side of the equation and move common variables together
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Example 5
Step 2: Group the x terms together and the y terms together
Step 3: Factor the GCF (coefficient)from the x group
and then from the y group
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Example 5
Step 4: Complete the square on the x group (don’t forget to multiply by the GCF before you add to the right side.)
Then do the same for the y terms
22=1 12=1
62=3 32=9
4(𝑥¿¿2+2𝑥+1)+9 ( 𝑦2+6 𝑦+9 )=36¿
9 ( 𝑦2+6 𝑦+9 ) +81
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Example 5
Step 5: Write the factored form for the groups.
**Now we have to make the equation equal 1 and that will give us our denominators
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Example 5 Step 6: Divide by the constant.
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Example 5 Step 7: simplify each fraction.
Now the equation looks like what we are used to!!
9 41
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(𝑥+1)2
9+
(𝑦+3 )2
4=1
What is the center of this ellipse?
What is the length of the major axis?
What is the length of the minor axis?
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Example 6: Ellipse
Step 2:
Step 1:
Step 3:
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Example 6
− 82=−4 −42=16 − 62=−3 −3 2=9
4(𝑥¿¿2−8𝑥+16)+25 (𝑦 2−6 𝑦+9 )=100¿
25 ( 𝑦2−6 𝑦+9 ) +225
4 (𝑥−4 )2+25 (𝑦−3 )2=100
Step 4:
Step 5:
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Example 6
25
41
Step 6:
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(𝑥−4)2
25+
(𝑦−3 )2
4=1
What is the center of this ellipse?
What is the length of the major axis?
What is the length of the minor axis?
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Example 7: Ellipse
Step 2:
Step 1:
Step 3:
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Example 7
42=2 22=4−
102 =−5 −52=25
9 (𝑥¿¿ 2+4 𝑥+4)+4 ( 𝑦2−10 𝑦+25 )=324 ¿
4 ( 𝑦2−10 𝑦+25 ) +100
9 (𝑥+2 )2+4 (𝑦−5 )2=324
Step 4:
Step 5:
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Example 7
36
811
Step 6:
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(𝑥+2)2
36+
(𝑦−5 )2
81=1
What is the center of this ellipse?
What is the length of the major axis?
What is the length of the minor axis?
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Example 8: Hyperbola
Step 2:
Step 1:
Step 3:
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Example 8
22=1 12=1
62=3 32=9
(𝑥¿¿2+2 𝑥+1)−9 ( 𝑦2+6 𝑦+9 )=18 ¿
−9 (𝑦2+6 𝑦+9 ) −81
(𝑥+1 )2−9 (𝑦+3 )2=18
Step 4:
Step 6:
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Example 8
21
Step 6:
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(𝑥+1)2
18− (𝑦+3 )2
2=1
What is the center of this hyperbola?
What is the length of the transverse axis?
What is the length of the conjugate axis?
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Example 9: Hyperbola
Step 2:
Step 1:
Step 3:
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Example 9
42=2 22=4 − 82=−4−42=16
4(𝑦¿¿ 2+4 𝑦+4)−9 (𝑥2−8 𝑥+16 )=36¿
−9 (𝑥2−8 𝑥+16 ) −144
4 (𝑦+2 )2−9 (𝑥−4 )2=36
Step 4:
Step 5:
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Example 9
9 41
Step 6:
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(𝑦+2)2
9− (𝑥−4 )2
4=1
What is the center of this hyperbola?
What is the length of the transverse axis?
What is the length of the conjugate axis?
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You Try! Write the equation of each conic section
in vertex form:
Identify the center of each conic section as well as the length of the major/minor or
transverse/conjugate axis.