Mrs. Turner s Precalculus page 0. Graphing Conics: Circles, Ellipses, Parabolas, Hyperbolas. Name: period:

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1 Mrs. Turner s Precalculus page 0 Graphing Conics: Circles, Ellipses, Parabolas, Hyperbolas Name: period:

2 9.0 Circles Notes Mrs. Turner s Precalculus page 1 The standard form of a circle is ( x h) ( y r The center is ( h, and the radius is r Find the center and the radius from the equations below: 1. x y 5. ( x 1) ( y 3) ( x 4) ( y 6) 18 Center: Center: Center: Radius: Radius: Radius: Graphing circles that are already in standard form Find the center and the radius of each circle. Then, graph it. 1. ( x 3) y 9. x ( y 5) 4 3. ( x 1) ( y 3) 9 Center: Center: Center: Radius: Radius: Radius: Graphing circles that are not in standard form. (Hint: We must complete the square!) STEP 1: Move the constant to the right STEP : Group x s and y s together STEP 3: Complete the square twice. Don t forget to add your magic number to the right hand side. STEP 4: Factor each trinomial. 1. x y 10 x 8y 16 0

3 . x y 4x 6y 1 Mrs. Turner s Precalculus page 3. x y x 4y 11 0 Write an equation for the circle that satisfies each set of conditions. 1. Center (8, -3); radius 6. Center (-5, ); passes through (-9, 6) 3. Center (3, 6); tangent to the x-axis 4. Endpoints of a diameter are (-4, -) and (8, 4) 5. Center (-4, -7); tangent to x =

4 9.1 Parabolas Notes Vertical Parabolas Horizontal Parabolas Equation ( x h) 4 p( y ( y 4 p( x h) Opens Up if 4p is positive Right if 4p is positive Down if 4p is negative Left if 4p is negative Vertex hk, hk, Focus ( h, k p) ( h p, Directrix y k p x h p Focal Width 4p 4p Axis of Symmetry x h y k The focus point will always be the parabola. The directrix line will always be the parabola. The vertex will always be halfway between the The focal width is the width of the parabola at the Mrs. Turner s Precalculus page 3 Graphing parabolas in standard form. 1. y 4x. ( x 1) 8( y 3) 3. ( y 4) ( x 5) 4. ( y 1) 4( x 3)

5 Mrs. Turner s Precalculus page 4 Graphing Parabolas NOT in standard form. (Hint: Complete the square!) 1. y 1y 1x 0. x y x Equation: Equation: 3. x 6x 4y 5 4. y 4y 0x 58 Equation: Equation:

6 Write the equation of the parabola that satisfies each set of conditions. Mrs. Turner s Precalculus page 5 1. Vertex (5, -1); Focus (3, -1). Vertex (1, 7); Directrix y = 3 3. Focus (3, 8); Directrix y = 4 4. Vertex (-7, 4); axis of symmetry x = -7; focal width = 6; 4p < 0 5. Vertex (4, 3); axis of symmetry y = 3; focal width = 4; 4p > 0

7 9. Ellipses Notes Mrs. Turner s Precalculus page 6 Major Axis Horizontal Vertical Standard Form ( x h) ( y ( x h) ( y 1 a b b a 1 Largest denominator goes with x Largest denominator goes with y Center ( h, ( h, Major Axis Vertices ( h a, ( h, k a) Minor Axis Vertices ( h, k b) ( h b, Foci ( h c, ( h, k c) c a b c Length of Major Axis a a Length of Minor Axis b b There are two foci for each ellipse. The foci are located the ellipse. Graphing ellipses in standard form. b a x y Center: Major V: Minor V: Foci: x y Center: Major V: Minor V: Foci: x 3. ( y 1) 1 16 Center: Major V: Minor V: Foci: x 1 ( y 3) Center: Major V: Minor V: Foci:

8 Graphing Ellipses that are NOT in standard form. Mrs. Turner s Precalculus page y x y 3. 9x 6y 36 x 1 y 1 Center: Major V: Minor V: Foci: Center: Major V: Minor V: Foci: 3. 4x 9y 40 x 36 y x 9y 100 x 18 y 116 Center: Major V: Minor V: Foci: Center: Major V: Minor V: Foci:

9 Write an equation for the ellipse which satisfies each set of conditions. Mrs. Turner s Precalculus page 8 1. Endpoints of major axis at (-11, 5) and (7, 5) Endpoints of minor axis at (-, 9) and (-, 1). Endpoints of major axis at (, 1) and (, -4) Endpoints of minor axis at (4, 4) and (0, 4) 3. Major axis is 0 units long and parallel to the y-axis Minor axis is 6 units long Center is (4, ) 4. Major axis is 16 units long and parallel to the x-axis Minor axis is 9 units long and parallel to the y-axis Center is (5, 4) 5. Endpoints of major axis at (10, ) and (-8, ) Foci at (6, ) and (-4, )

10 9.3 Hyperbolas Notes Horizontal Vertical Standard Form ( x h) ( y ( y 1 a b a Center ( h, ( h, Foci ( h c, ( h, k c) ( x h) b c a b c a b Vertices ( h a, ( h, k a) Asymptotes b a y k ( x h) y k ( x h) a b 1 Mrs. Turner s Precalculus page 9 The transverse axis is the segment whose endpoints are the vertices of the hyperbola The conjugate axis is the segment that is perpendicular to the transverse axis Graphing a hyperbola in standard form. x y y x Foci: Foci: ( x ) 3. ( y 3) ( x 3) 144 ( y ) 5 Foci: Foci:

11 Graphing a hyperbola NOT in standard form. Mrs. Turner s Precalculus page x y 16. 6y 4x 36 y 8x 6 Foci: Foci: 3. 4x 3y 8x x 4y 54 x 40 y 55 0 Foci: Foci:

12 Write an equation for the hyperbola that satisfies each set of conditions. Mrs. Turner s Precalculus page Vertices (-5, 0) and (5, 0) Conjugate axis of length 1. Vertices (0, -4) and (0, 4) Conjugate axis of length Vertices (9, -3) and (-5, -3) Foci 53, 3 4. Vertices (-4, 1) and (-4, 9) Foci 4, Centered at the origin Horizontal Transverse axis of length 8 Conjugate axis of length 6

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