Translation and Rotation of Axes

1
Translation and Rotation of Axes

• 8.4

2
Solving Second Degree Equations in Two Variables

• Use the quadratic formula to solve for y.
• a will be the coefficient of the y2 term.
• b will be the coefficient of the y term.
• c will be the constant term and the terms with
  xs in them.
• The coefficients may contain xs.
• Just solve for y.

3
Translation of Axes Formulas

• The coordinates (x,y) and (x, y) based on
  parallel sets of axes are related by either of
  the following translation formulas
• x x h y y h
• or
• x x h y y - k

4
Completing the Square to Find a Conic Equation

• Put the terms with variables in decreasing order
  on the left side and the constants on the right
  side of the equation.
• Factor out the coefficient of the y2 term from
  the y2 and the y terms. Call this m.
• In the parenthesis, add (b/2)2.
• Now add m(b/2)2 to each side of the equation.

5
Completing the Square to Find a Conic Equation

• Repeat for the x terms.
• Divide both sides of the equation by the constant
  if you have a hyperbola or ellipse.
• For a parabola, you will only need to complete
  the square once.

6
Rotation of Axes Formula

• The coordinates (x,y) and (x, y) based on
  rotated sets of axes are related by either of the
  following rotation formulas
• x x cos a y sin a
• y -x sin a y cos a
• or
• x x cos a y sin a
• y x sin a y cos a
• Where a, 0ltalt pi/2, is the angle of rotation.
  Example 10 from 7.2 proves the first set of
  equations.

7
Coefficients for a Conic in a Rotated System

• See page 670

8
Angle of Rotation to Eliminate the Cross Product
Term

• If B ltgt 0, an angle of rotation a such that
• Cot 2a (A C)/B and 0ltaltpi/2
• Will eliminate the term Bxy from the second
  degree equation in the rotated xy coordinate
  system.

9
Angle of Rotation to Eliminate the Cross Product
Term

• If B ltgt 0, an angle of rotation a such that
• Cot 2a (A C)/B and 0ltaltpi/2
• Will eliminate the term Bxy from the second
  degree equation in the rotated xy coordinate
  system.

10
Discriminant Test

• The second degree equation Ax2 Bxy Cy2 Dx
  Ey F 0 graphs as
• A hyperbola if B2 4AC gt 0
• A parabola is B2 4AC 0
• An ellipse if B2 4AC lt 0
• Except for degenerate cases. Pg. 631 figure 8.2 b