Precalculus Conic Sections Unit 6. Parabolas. Label the parts: Focus Vertex Axis of symmetry Focal Diameter Directrix

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1 PICTURE: Parabolas Name Hr Label the parts: Focus Vertex Axis of symmetry Focal Diameter Directrix Using what you know about transformations, label the purpose of each constant: y a x h 2 k It is common to rewrite the equation of the parabola in standard form: 2 y k 4p x h 2 x h 4p y k Vertex: Focus: Vertex: Focus: p > 0 p < 0 p > 0 p < 0

2 Examples: Describe the graph (y 1) 2 = 4(x 1). Write the equation for the parabola. Write the equation for the parabola. Find the equation with F( 3, 4), directrix y = 2.

3 Graph the parabola: y y = x + 1 Find the equation with V( 2, 3) and F(0, 3). Graph the parabola: x 2 + 6x 4y + 1 = 0. Engineers often design roads with parabolic surfaces to allow rain to drain off. A particular road that is 32 feet wide is 0.4 foot higher in the center than on the sides. a) Find the equation of the parabola. (Place the vertex at the origin.) b) How far from the center of the road is the road surface 0.1 foot lower than in the middle?

4 Ellipses PICTURE: x h y k a 1 b Label the parts: Center Major Axis Minor Axis Vertices Foci Eccenticity: EXAMPLES: Find the equation of the ellipse.

5 Graph the ellipse: (x 2) 2 4 (y + 3)2 + = 1 8 Graph the ellipse: x 1 y Find an equation with C(2, 3), one focus (3, 3) and one vertex (5, 3) Find the equation with V(±4, 0) and F(±2, 0)

6 Graph the ellipse: 4x 2 + y 2 8x + 4y + 4 = 0 Graph the ellipse: 3x 2 + 5y 2 12x + 30y + 42 = 0 Where is this applied? Newton s work showed that objects in closed orbits must have circular or elliptical paths. However, if the velocity of an orbiting body is increased, its orbital path changes to a parabola or hyperbola, and it escapes the gravitational pull of the Sun and leaves the solar system. APPLICATION: Halley s Comet passed by Earth in has an elliptical orbit with the sun at one focus. The eccentricity of the orbit is The length of the major axis of the orbit is approximately astronomical units. (An astronomical unit is about 93 million miles.) Find the standard form of the equation of the orbit. Place the center of the orbit at the origin and place the major axis on the x-axis. (NOTE: You will have a chance to view Halley s Comet in 2061 when is passes by Earth again.)

7 Hyperbolas PICTURE: x h y k a 1 b y k x h b 1 a Label the parts: Center Foci Transverse Axis Minor Axis Vertices Asymptotes EXAMPLES: Find the equation of the hyperbola.

8 Graph the hyperbola: (x + 2) 2 9 (y 5)2 = 1 49 Determine the equation of the hyperbola having center C( 1, 1), a vertex on the y-axis, and a focus at F( 3, 1). Graph the hyperbola: x 2 4x 4y 2 8y = 1 Graph the hyperbola: 4x 2 3y 2 + 8x + 16 = 0 Find the equation for the hyperbola with F (0, ±3), transverse axis length 4.

9 CLASSIFYING CONICS: Ax 2 + Cy 2 + Dx + Ey + F = 0 START Is A or C equal to 0? YES PARABOLA NO Is A = C? YES CIRCLE NO Do A and C have the same sign? ELLIPSE HYPERBOLA

10 PRACTICE: Write an equation in standard form for each conic section Focus Focus Find the equation of the conic section in standard form with the given characteristics: 5. HYPERBOLA with vertices (0, ±2) and asymptotes y = ±2x 6. PARABOLA with vertex (3, 3) and focus (3, 9 4 ) 7. ELLIPSE with foci (±2, 0) and major axis length of 10

11 Classify each conic section, write in standard form, and graph (including foci) x 10x y x y x y y x 20y x y x 2 8 0

12 5. 2 y x y x 25y 24x 250 y x 4y 54x 8y x 25y 100 y

13 9. Stein Glass Co. makes parabolic headlights for a variety of automobiles. If one of its headlights has a parabolic surface generated by the parabola x 2 = 12y where should its light bulb be placed? 10. A cannon fires a cannonball. The path of the cannonball is parabolic with vertex at the highest point of the path. If the cannonball lands 1600 feet from the cannon and the highest point it reaches is 3200 feet above the ground, find an equation for the path of the cannonball. (HINT: Place the origin at the location of the cannon.) 11. A lithotripter machine uses an elliptical reflector to break up kidney stones. A spark plug in the reflector generates energy waves at one focus of an ellipse. The reflector directs these waves toward the kidney stone positioned at the other focus of the ellipse with enough energy to break up the stone. The lengths of the major and minor axes of the ellipse are 280 mm and 160 mm. How far is the spark from the kidney stone? Spark Plug Kidney Stone

14 x h y k 12. The equation a describes a degenerate ellipse. What does b 0 this mean and how is it different from a regular ellipse? If you need a hint, go to the website and click on the Degenerate Conics link. TRUE OR FALSE??? Explain your answers. 13. It is easier to distinguish the graph of an ellipse from the graph of a circle when the eccentricity of the ellipse is close to The area of a circle with diameter d = 2r = 8 is greater than the area of an ellipse with major axis 2a = If the vertex and focus of a parabola are on a horizontal line, then the directrix of the parabola is a vertical line. 16. If the asymptotes of the hyperbola a = b. x a y b 1 intersect at right angles, then

15 Complete the square (Part 1) 1. y = x 2 + 2x y = x 2 20x y = 2x x 4 4. y = 3x x x = 3y 2 9y x = 5y y + 8 Complete the square (Part 2) 1. x 2 + 4y 2 + 6x 8y + 9 = x 2 + y 2 8x + 4y 8 = x 2 4y 2 54x + 40y + 37 = x 2 9y x 36y + 43 = 0

16 Trig Tuesday: Simplify the following expressions and match them with their equivalent term. (You will not use all the letters in the right-hand column.) You must show work to receive credit. 1. sec 2 x 1 A sin x 3. sin x cot x 4. 2 sin x 1 cos x tan 2 x 6. csc 2 x cot 2 x B. 2 C. 1 D. csc x E. sec x + csc x F. cot 2 x G. sin x cos x 1 sin x sin x cos x sin xcos x 2 cot x csc x 1 H. tan 2 x I. sec x + 1 J. cos x K. sec 2 x L cos x cot x 11. sin 2 x + cos 2 x + 1 M. 1 sin x N. csc x + 1 O. 1 cos x