Pre-Calculus Accelerated Name: Conics and Parametrics ...

Review Sheet Conics/Parametrics 4/27/07

Pre-Calculus Accelerated Name: _______________________________ Conics and Parametrics Period: _________ Date: ________________ Review Sheet 2007

Due the day of the test. All work must be done NEATLY in pencil. Show all work! Calculator may be used on * problems only!!!

Conic Sections When graphing, include the following: Circle: center, radius, domain and range Ellipse: center, vertices (all 4), domain and range Hyperbola: center, vertices (2), draw the box & asymptotes, write the equations of asymptotes, domain and range Parabola: vertex, intercept(s), axis of symmetry, domain and range 1. Graph the following: x

2+ y

2! 12x + 10y + 45 = 0

Domain: Range: 2. Graph the following: x = !2( y + 2 )

2+ 6

Domain: Range:


3. Graph the following. 4 x

2+ 6 y

2! 16x + 12 y ! 2 = 0

Domain: Range: 4. Graph the following. 6.25 x

2! y

2= 25

Domain: Range:

5. Find the points of intersection: 4 x

2+ 4 y

2=25

2x + y2= 1

!"#

$#


For problems 6-8, identify the type of conic section (circle, ellipse, hyperbola, or parabola) and write the equation of the conic section which fits each description. 6. A center at ( - 2, 7 ) , a vertex at ( - 2, 14 ), and a minor axis with a length of 4. Type:___________________ Equation:____________________________ 7. A center at ( 3, 5 ), tangent to the x-axis at ( 3, 0 ) and contains the point ( 6, 1 ). Type:___________________ Equation:____________________________

8. An x - intercept at ( 3, 0 ) and an asymptote with the equation y =7

3x .

Type:___________________ Equation:____________________________ 9. Write the equation of the ellipse: Equation:_____________________________ 10. Write the equation of the tangent line to ( x ! 3) 2 + ( y ! 4 ) 2 = 5 through the point P ( 4,2 ) .

�


11. Given the table below, sketch the following graphs. Then write the rectangular and parametric equations.

12. Eliminate the parameter and sketch the rectangular graph given the parametric equations. State any restrictions that exist on x, y and t.

a. x = t

2+ 3

y = t +1

!"#

b. x = t

3 ! 4

y = 5 ! t 3"#$

%$

c. x = sin t

y = 2cos t

!"#

d. x =

1

t

y = t + 2

!

"#

$# hint: rational function

t

(x, y)

0

(6, 0)

3

(5, 2)

6

(4, 0)

9

(3, -2)

12

(2, 0)

15

(1, 2)

18

(0, 0)

y

x

x

t

y

t


*13. Determine if the following will have a simultaneous solution. If they do, find when and where they meet. If not, explain how you know.

a. x1= t !15

y1= t + 3

"#$

and x2= !t +1

y2= 2t ! 5

"#$

b. x1= t

2

+ 2

y1= t

3 !1

"#$%

and x2= !2t +1

y2= t

2 !1

"#$

c. x1= 3cos t

y1= 3sin t

!"#

and x2= 2t ! "

y2= t +1.43

#$%

14. Given the rectangular graph, write the corresponding parametric equations.

10

8

6

4

2

-2

- 4

- 6

- 8

-15 -10 - 5 5 10 15

(3, 10) t = 8

(2, 3) t = 6 (1, 2) t = 4

(0, 1) t = 2

(-1, -6) t = 0


15. Given the rectangular graph, write the corresponding parametric equations. 16. Given the parametric graphs below, write the rectangular equation. *17. A ferris wheel with radius of 24 feet is rotating counterclockwise at a rate of one revolution every 90 seconds. Bob is getting on the ride at t = 0, 8 feet off the ground. A bird takes off at the same moment from a spot on the ground that is 100 feet to the right of the center of the platform and flies straight towards the ride. After 10 seconds, the bird has flown to a point 25.34 feet left of where he started and 14.618 feet off the ground. Will Bob and the bird collide if the bird continues on the same path? a. Sketch a picture. b. Write the parametric equations for Bob and the bird. c. Determine if Bob and the bird will collide. If yes, give time and location. If no, justify your answer.

6

5

4

3

2

1

-1

- 2

- 6 - 4 - 2 2 4 6

t=0 @ (4, 3)

t=5 @ (1, 4)

t=10 @ (-2, 3)

t=15 @ (1, 2)

8

6

4

2

-5 5

3

2

1

-1

- 2

- 3

- 4

- 5

- 6 - 4 - 2 2 4 6

t

t

y x


Review:

18. a. Convert 160° to radians. ____________ b. Convert 7!

5 to degrees. ____________

19. Find the reference angle for each of the following:

a. 12!

7 b. 500°

20. Sketch one cycle of the following: y = ! sec 2x( ) !1 amplitude: __________

period: _____________

phase shift: _________

vertical shift: ________

domain: ____________

range: ______________

Even, Odd, or Neither? 21. Solve for x over the given interval :

a. sin2x + 3cos

2x = 2 !" ,"[ ) b. tan 2x +

!3

"#$

%&'= 3 0,2![ )

22. Given sec! = "25

7 and sin ! > 0 , find sin ! +

7"6

#$%

&'(

.


SOLUTIONS: 1.) Center ( 6, -5 ) radius = 4 Domain: 2, 10!" #$

Range: !9, !1[ ] 2.) Vertex; ( 6, - 2 ) Domain: !",6( ]

Range: !" , "( ) x-intercept: ( - 2, 0 ) axis of symmetry: y = -2 3.) Center ( 2, -1 )

Vertices: (2, 1), (2, -3), 2! 6 , !1( ) and 2+ 6 , !1( )

Domain: 2! 6 , 2+ 6"#

$%

Range: ! 3, 1[ ] 4.) Center: (0, 0) Vertices: (2, 0) and (-2, 0)

equation of the asymptotes: y = ±5

2x

Domain: !",!2( ]# 2, "[ )

Range: !" , "( )

5.) !3

2, 2

"#$

%&'and !

3

2, ! 2"

#$%&'

6.) Ellipse ( x + 2 )

2

4+( y ! 7 )

2

49= 1

7.) Circle: ( x ! 3) 2 + ( y !5 )2= 25

8.) Hyperbola x2

9!

y2

49= 1

9.) ( x ! 1)

2

9+( y + 1)

2

4= 1

10.) y=1

2x or x ! 2 y = 0

11.) rect: y=2sin!2x

"#$

%&'

para: x=!1

3t+6, y=2sin

"6t

#$%

&'(

12.) a. x= y!1( )2 +3 , x > 3 (sideways parabola) b. y = -x + 1 (line)

c. x2 + y2

4=1 (ellipse)

d. y= 1x+2 or y=

1+2x

x (rational function)

t!0, x!0, y!2 13.) a. yes, when t = 8, they intersect at (-7, 11). b. no, show work or explain your reasoning c. yes, when t = 1.57, they intersect at (0, 3)

14. x= 12t!1!!!!and !!!y=

1

8t!4( )3+2

15. x = 3cos!10t

"#$

%&' + 1!!!!!!!and !!!!!!!!y = sin

!10t

"#$

%&' + 3

16. (x!4)2

9+(y+1)

2

9=1

17. The bird and Bob DO collide when t = 32 seconds.

18 a.) 8!

9

b.) 252

�

°

19. a) 2!

7

b) 40

�

° 20. amp of cos = 1; period = π; ps=none; vs=down 1;

D: ! except ..."

4,

3"

4... ; R: !",!2( ]# 0,"[ ) ; even

21. a) {π/4, 3π/4, -3π/4, -π/4} b) {0, π/2, π, 3π/2}

22. 7!24 3

50

Pre-Calculus Accelerated Name: Conics and Parametrics ...

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