1. SECTION: 10 – 4 ROTATIONS WARM-UP Find the center, vertices, foci, and the equations of the asymptotes of each hyperbola. 3. Write the standard form of the hyperbola with vertices (–10,3) and (6,3) and foci (–12,3) and (8,3).

2. DISCRIMINANT. Given the equation Ax 2 +Bxy+Cy 2 +Dx+Ey+F=0, the quantity B 2 –4AC is the discriminant. CLASSIFICATION OF CONIC SECTIONS BY THE DISCRIMINANT 1. If B 2 –4AC<0, then the graph of the equation is either a circle or an ellipse. 2. If B 2 –4AC=0, then the graph of the equation is a parabola. 3. If B 2 –4AC>0, then the graph of the equation is a hyperbola.

3. ROTATED CONIC SECTIONS. Some conic sections may be rotated so that they are not parallel to either the x- or the y-axis. GENERAL FORM OF THE EQUATION. The general form of the equation of a rotated conic section is Ax 2 +Bxy+Cy 2 +Dx+Ey+F=0. Notice the Bxy term contains both the x- and y-variables. This equation is also referred to as the equation in the xy-plane.

4. ROTATION OF AXES TO ELIMINATE AN xy- TERM. Rotation of the axes is the process used to eliminate the xy-term in the general form of the equation. The objective is to rotate x- and y-axes until they are parallel to the axes of the conic section. The rotated axes are denoted as the x’-axis and the y’-axis.

5. θ x’ y’

6. FORMAL DEFINITION. The general second- degree equation Ax 2 +Bxy+Cy 2 +Dx+Ey+F=0 can be rewritten as: A’(x’) 2 +C’(y’) 2 +D’x’+E’y’+F’=0 by rotating the coordinate axes through an angle θ, where The coefficients of the new equation are obtained by making the substitutions

7. ELIMINATING THE xy-TERM 1. Identify the A, B, and C values. 2. Determine the angle measure of the rotation using the formula 3. Find the value of using the half- angle formulas.

8. 4. Write the rotated equation by substituting into the original equation and simplify.

9. EXAMPLE 1. Determine the type of conic section. Rotate the conic section to eliminate the xy-term. Then write the standard form of the equation.

13. d. Transform the equation to an x’y’ equation by a rotation of 45°.

14. CLASS WORK/HOMEWORK: SECTION: 10 – 4 PAGE: 729 PROBLEMS: 5 – 15 ODD (DO NOT SKETCH)