PreCalculus Honors Name ____________________________ Period______ Summer Packet 2014–15 Welcome to PreCalculus Honors! In this course you will be presented with mathematical tasks that require the application of previous course work in new and unfamiliar situations. The problems in this packet were selected to provide a sampling of concepts, skills, and solution methods with which you have experience from previous courses. Complete this packet on your own. However, you may collaborate with your peers, but be aware of your area(s) of weakness and take time to fully review the topic(s) (versus simply completing the problem.) You may find these websites useful: http://www.purplemath.com/modules/index.htm or http://www.khanacademy.org/#browse There will be a brief amount of time to get help on the first day of class in August. On the second day of class, this packet will be collected and you will take a quiz covering this material. The purpose of this is to provide feedback on your understanding of these topics. This packet should be completed without the use of a calculator unless otherwise stated. Use your calculator as a tool to verify answers when applicable. Show all work needed to arrive at your answer. This packet can also be found on the math department website at: http://www.d125.org/academics/mathematics_summer_packets.aspx 1. Simplify to a single fraction:

a.

1x + 3

+ 1x

x b. x

2 + 7 x + 12x 2 − 16

2. Given the line 3x – 5y = 7, find the point-slope form of the equation of a line through 3, 1( ) that is a. parallel to the given line b. perpendicular to the given line 3. Given f (x) = 2 x + 5, find: a. f (x + 3) b. f (f (x)) c. f (x + 1) – f (x)

4. a. The following are graphs that you need to be able to graph quickly and accurately without a calculator. They should be memorized. Sketch each accurately on the graphs below – label at least two points. G1: y = x G2: y = c (c is a constant) G3: y = x2 G4: y = x3 G5: y = |x| G6: y = [x] (y is the greatest integer ≤ x) G7: y = x G8: y = x3 For G9 and G10: Draw asymptotes as a dotted line and label at least two points on the graph. G9: y = ex G10: y = ln x

4b. The following are graphs with transformations applied. Graph accurately labeling asymptote (if applicable) and at least two points. Think about the shifts and reflections that are happening from the original graph. G11: f x( ) = − x + 3 G12: g x( ) = ex + 2 5. Factor completely.

a. x − 5( ) 2 − y 2 b. x 3 − 64 c. 3x3 − 6 x 2 − 45 x (HINT difference (HINT difference of squares) of cubes) 6. Solve for all solutions (real and non-real) (you may use results from problem #5). a. x − 5( ) 2 = 9 b. x 3 − 64 = 0 c. 3x3 − 6 x 2 − 45 x = 0 7. Find the quotient and remainder when x3 − 6 x 2 − 5 x − 7 is divided by x – 5. 8. Given f (x) = − x 2 + 6 x + 7 , find each of the following: a. the coordinates of the vertex b. the x and y intercepts

9. A parabola has a vertex at (5, 2), opens down and contains the point (1, –6). a. Write the equation of the parabola in conic form ( 4 p y − k( ) = x − h( )2 where (h, k) is the vertex). b. Sketch the graph using the vertex and the endpoints of the latus rectum (4p = focal diameter).

c. Write the equation in vertex form ( y = 14 p

x − h( )2 + k ).

10. Solve each inequality - give your solution in interval notation and graph on a number line. a. x2 + 2x −15 ≥ 0 b. 2 x + 3 < 7 11. Solve each for x.

a. 14

= 8 x + 3 b. 27 x + 1 = 9 2 x − 4

12. Solve for x.

a. log 3 9 = x b. log x 8 =32

c. loga x = 3 d. ln e x = 4 13. Given y = 2 + log 3 x − 1( ) , a. find the domain. c. Sketch the graph showing the asymptote and at least one point.

b. find the x intercept(s). 14. Solve each for x. a. log x + 2( ) 16 = 4 b. log 3 x + log 3 (x − 2) = 1 15. In each, find the length of segment AB. Calculator allowed. Round to nearest tenth. a. b.

A

C

B

12 cm

67°

83°

A

C

B

6 cm

8 cm 37°

16. Find the sum:

a. n2

n − 2n=3

6

∑ b. 8 12

⎛⎝⎜

⎞⎠⎟n

n=0

∞

∑

17. Given the sequence : 2 x − 3, 7 x, 11x ; find x if the sequence is

a. arithmetic b. geometric (x ≠ 0) 18. Find the indicated trig function and the value of θ (rounded to the nearest tenth of a degree). Calculator allowed to find θ . a. b. cosθ = _________ sinθ = __________ tanθ = _________ secθ = __________ cscθ = _________ cot θ = _________ θ = __________ θ = __________ 19. Find the missing sides in each. Give exact answers. a. b.

45°

x 6

y

30°

y

6

x

12

16θ

6

3

θ

20. Find the point(s) of intersection. Show all work.

a. 3x − 3y = 9x + 4y = −22

b. x2 + y2 = 4y = 2 − x2

21. Given y = x2 + 7 x + 12x 2 − 16

, find the following:

a. x intercept(s) b. y intercept c. equation(s) of vertical asymptote d. equation of horizontal asymptote e. location of hole(s) f. sketch the graph showing all key points 22. Solve the matrix equation for x and y (answers on the back are written as (x, y)):

2x x − 33 y

⎡

⎣⎢⎢

⎤

⎦⎥⎥i x

−4⎡

⎣⎢

⎤

⎦⎥ +

23

⎡

⎣⎢

⎤

⎦⎥ =

20−20

⎡

⎣⎢

⎤

⎦⎥

y

x

Answers

1 a. 2x + 3x3 + 3x2

b. x + 3x − 4

2 a. y −1 = 35x − 3( ) b. y −1 = −5

3x − 3( )

3 a. 2x+11 b. 4x+15 c. 2 4. Check using your calculator 5a. (x − 5 + y)(x − 5 − y) b. (x − 4)(x2 + 4x +16) c. 3x(x − 5)(x + 3) 6a. x ∈ 2,8{ } b. x ∈ 4,−2 + 2i 3,−2 − 2i 3{ } c. x ∈ −3,0,5{ } 7. Quotient: x2 − x −10 Remainder: −57 8 a. (3, 16) b. x-ints: –1 and 7 y-int: 7

9. a. −2 y − 2( ) = x − 5( )2 b. endpoints of latus rectum are (4, 1.5) and (6, 1.5) c. y = − 1

2x −5( )2

+ 2

10 a. x ∈ (−∞,−5]∪ [3,∞){ } b. x ∈ −5,2( ){ }

11 a. x ∈ −113

⎧⎨⎩

⎫⎬⎭

b. x ∈ 11{ }

12 a. 2 b. 4 c. a3 d. 4

13 a. (1,∞) b. 109

14 a. 0 b. 3 15 a. 4.8 cm b. 22.1 cm

16 a. 1033

b. 16

17 a. –3 b. −119

18 a. cscθ =53; θ = 36.9° b. cot θ =

111

; θ = 73.2°

19 a. x = 6 2 y = 6 b. x = 4 3 y = 2 3 20. a. (–2, –5) b. 0,2( ), 3,−1( ), − 3,−1( ) 21 a. –3 b.

− 3

4 c. x=4 d. y=1 e.

−4,1

8⎛⎝⎜

⎞⎠⎟

22. 3,8( ) and   −1,5( )