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Five-Minute Check (over Lesson 7-3)
Then/Now
Key Concept:Rotation of Axes of Conics
Example 1:Write an Equation in the x′y′-Plane
Key Concept:Angle of Rotation Used to Eliminate xy-Term
Example 2:Write an Equation in Standard Form
Key Concept:Rotation of Axes of Conics
Example 3:Real World Example: Write an Equation in the xy-Plane
Example 4:Graph a Conic Using Rotations
Example 5:Graph a Conic in Standard Form
Over Lesson 7-3
A. B.
C. D.
Over Lesson 7-3
A. B.
C. D.
Over Lesson 7-3
Graph the hyperbola 4x2 – y2 + 32x + 6y + 39 = 0.
A. B.
C. D.
Over Lesson 7-3
Write an equation for the hyperbola with foci (10, –2) and (–2, –2) and transverse axis length 8.
A.
B.
C.
D.
Over Lesson 7-3
Determine the eccentricity of the hyperbola given by 9y2 – 4x2 – 18y + 24x – 63 = 0.
A. 0.555
B. 0.745
C. 1.180
D. 1.803
You identified and graphed conic sections. (Lessons 7–1 through 7–3)
• Find rotation of axes to write equations of rotated conic sections.
• Graph rotated conic sections.
Use θ = 90° to write x 2 + 3xy – y
2 = 3 in the x y -plane. Then identify the conic.
Find the equations for x and y.
Write an Equation in the x y -Plane
= –y
x = x cos θ – y sin θ Rotation equationsfor x and y
y = x sin θ + y cos θ
sin 90 = 1 and cos 90 = 0
= x
Substitute into the original equation.
x 2 + 3xy – y
2 = 3
(–y )2 + 3(–y )(x ) + (x ) 2 = 3
(y )2 – 3x y + (x ) 2 = 3
Write an Equation in the x y -Plane
Answer:
Write an Equation in the x y -Plane
Use θ = 60° to write 4x 2 + 6xy + 9y
2 = 12 in the x y -plane. Then identify the conic.
A.
B.
C.
D.
Rotation of the axes
Write an Equation in Standard Form
Using a suitable angle of rotation for the conic with equation x
2 – 4xy – 2y 2 – 6 = 0, write the equation in
standard form.
The conic is a hyperbola because B2 – 4AC > 0. Find θ.
A = 1, B = –4, and C = –2
Write an Equation in Standard Form
–3
Write an Equation in Standard Form
Use the half-angle identities to determine sin θ and cos θ.
Half-Angle Identities
Simplify.
Write an Equation in Standard Form
Next, find the equations for x and y.
Rotation equations for
x and y
Simplify.
Write an Equation in Standard Form
Substitute these values into the original equation.
x2 – 4xy – 2y2 = 6
Write an Equation in Standard Form
Answer:
A.
B.
C.
D.
Write an Equation in the xy-Plane
Use the rotation formulas for x and y to find the equation of the rotated conic in the xy-plane.
Write an Equation in the xy-Plane
Substitute these values into the original equation.
= x cos 45° + y sin 45° θ = 45° = y cos 45° – x sin 45°
Rotation equations for
x′ and y′
Write an Equation in the xy-Plane
Original equation
Multiply each side by 16.
Substitute.
Simplify.
2(x′)2 + (y′)2 = 16
Write an Equation in the xy-Plane
Answer: 3x2 + 2xy + 3y2 – 32 = 0
Combine like terms.
Simplify.
A.
B.
C.
D.
ASTRONOMY A sensor on a satellite is modeled by
after a 60° rotation. Find the equation
for the sensor in the xy-plane.
The equation represents an ellipse in standard form. Use the center (0, 0), vertices (–6, 0), (6, 0), and co-vertices (0, –3) and (0, 3) in the x′y′-plane to determine the corresponding points for the ellipse in the xy-plane.
Graph a Conic Using Rotations
Find the equations for x and y for = 60°.
Graph a Conic Using Rotations
x = x cos – y sin Rotation equations y = x sin + y cos for x and y
Use the equations to convert the xy-coordinates of the vertex into xy-coordinates.
Graph a Conic Using Rotations
= –3
Graph a Conic Using Rotations
Graph a Conic Using Rotations
The new vertices and co-vertices can be used to sketch the ellipse. They can also be used to identify the x′y′-axis.
Answer:
Graph a Conic Using Rotations
A. B.
C. D.
Use a graphing calculator to graph the conic section given by 8x2 + 5xy – 4y2 = –2.
8x2 + 5xy – 4y2 = –2 Original equation
8x2 + 5xy – 4y2 + 2 = 0 Add 2 to each side.
–4y2 + (5x)y + (8x2 + 2) = 0 y-terms inquadratic form
Graph a Conic in Standard Form
Quadratic formula
Multiply.
Graph a Conic in Standard Form
Simplify.
Graphing both equations on the same screen yields the hyperbola.
Answer:
Use a graphing calculator to graph the conic section given by 3x2 – 6xy + 8y2 + 4x – 2y = 0.
A. B.
C. D.
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