Northwest High School’s Precalculus/Honors Precalculus

Summer Review Packet

This packet is optional! It will NOT be collected for a grade next school year.

This packet has been designed to help you review various mathematical topics that are necessary for your success in precalculus next year.

Instructions:

DO ALL PROBLEMS WITHOUT USING A CALCULATOR J

If you have any questions regarding the summer math packet, please contact Debbie Wilson by email Deborah_J_Wilson@mcpsmd.org.

Khan Academy is another great resource that will show you videos on all the topics listed in this packet and much more! You can check it out at the link below:

https://www.khanacademy.org/math/algebra2ENJOY YOUR SUMMER! WE ARE LOOKING FORWARD TO SEEING YOU IN THE FALL.

I.RewritingandSolvingEquations

A. Solveeachequationfory.Example:3𝑦 + 2𝑥 = 53𝑦 = −2𝑥 + 5𝑦 = − !

!𝑥 + !

!

1. 7𝑦 + 6𝑥 = 10 2. 4𝑥 − 6𝑦 = 2

3.2𝑥 − 𝑦 = −8 4.5 − 2𝑥 = 9𝑦 + 2

B. Solveeachquadraticequationforx.Example:7x 3𝑥 + 2 𝑥 − 5 = 0 7x=0 3𝑥 + 2 = 0 and 𝑥 − 5 = 0 Seteachfactorequaltozeroandsolve X=0 3𝑥 = −2 𝑥 = 5 𝑥 = − !

!

5. 𝑥 − 1 5𝑥 + 3 = 0 6.2𝑥 𝑥 − 4 = 0

7.2𝑥! = 32 8.𝑥 𝑥 − 3 = 0

C.Factoringandsolvingquadraticequationsoftheform𝑨𝒙𝟐 + 𝑩𝒙 + 𝑪 = 𝟎,𝒘𝒉𝒆𝒓𝒆 𝑨 = 𝟏

Example:𝑥! + 7𝑥 + 6 = 0

𝑥 + 1 𝑥 + 6 = 0Think:Whatmultipliesto6andaddsto7?Answer:1and6

𝑥 + 1 = 0and 𝑥 + 6 = 0

𝑥 = −1 𝑎𝑛𝑑 𝑥 = −6

9.𝑥! + 3𝑥 − 4 = 0 10.𝑥! − 4𝑥 − 21 = 0 11.𝑥! + 19𝑥 + 90 = 0

D.Factoringandsolvingquadraticequationsoftheform𝑨𝒙𝟐 + 𝑩𝒙 + 𝑪 = 𝟎,𝒘𝒉𝒆𝒓𝒆 when𝑨 ≠ 𝟏

Example:2𝑥! + 7𝑥 + 3 = 0 Startwiththeequationintheform𝐴𝑥! + 𝐵𝑥 + 𝐶 = 0.

Factorthegreatestcommonfactor(GCF),ifpossible.

Thinkofwhatmultipliesto“a●c”andaddsto“b”?(Finda●c=2●3=6).

Ask,whatmultipliesto“6”andaddsto“7”?(Answer:1and6)

2𝑥! + 𝟔𝒙 + 𝟏𝒙 + 3 = 0 Write1and6asyournewmiddlelinear(“x”)terms.

2x(x+3)+1(x+3)=0 Factorbygrouping.

(2x+1)(x+3)=0 Seteachfactorequaltozeroandsolve.

𝑥 = − !!and 𝑥 = −3

12.2𝑥! + 𝑥 − 6 = 0 13.2𝑥! − 4𝑥 − 16 = 0 14.7𝑥! + 35𝑥 − 42 = 0

E.For each of the following functions:

a) Graph the function b) State the domain of the function using interval notation. Example: [− 3, ∞) or (− 2, 7) c) State the range of the function using interval notation

15.𝑓 𝑥 = − !

!𝑥 + 4 16.𝑓 𝑥 = 3𝑥 + 2 17.(𝑥 + 2)! + 1

18.𝑓 𝑥 = 𝑥! + 6𝑥 + 1 19.𝑓 𝑥 = 𝑥 + 3 20.𝑓 𝑥 = 𝑥

21.𝑓 𝑥 = 𝑥 + 2 22.𝑓 𝑥 = 𝑥 − 4 23.𝑓 𝑥 = !

!!!

II.ParentFunctionsandtheirProperties

24.Identifythefunctionsaslinear,quadratic,cubic,orabsolutevalue.

a.) b.)

Function: ___________________ Function: ___________________

c.) d.)

Function: ___________________ Function: ___________________

25. Given the graphs below, state the domain and range. (Remember, the domain is the x-values and the range is the y-values.) a.)

a.) Domain: _____________ a.) Domain: _____________ a.) Domain: ____________

b.) Range: ______________ b.) Range: ______________ b.) Range: _____________

26. (Multiple Choice) To the bottom left is the graph of 𝑦 = 𝑥!. Use the parent function and your knowledge of

b.) c.)

transformations to determine which equation could represent the graphs to the right.

Parent function

a.) 𝑦 = −4𝑥! a.) 𝑦 = −4𝑥! b.) 𝑦 = 𝑥! + 4 b.) 𝑦 = 𝑥! + 4 c.) 𝑦 = 𝑥! − 4 c.) 𝑦 = 𝑥! − 4 d.) 𝑦 = (𝑥 − 4)! d.) 𝑦 = (𝑥 − 4)!

III. Pythagorean Theorem & Trigonometry

A. Solve for the missing side of the triangle using the Pythagorean Theorem:

Recall: Pythagorean Theorem: 𝑎! + 𝑏! = 𝑐! 27. a = 6 ft. b = 8 ft.

28. b = 7 m. c = 9 m. 29. a = 5 cm. c = 13 cm. 30. a = 4 in. b = 8 in.

B. Using SOHCAHTOA from Geometry

where 𝑠𝑖𝑛𝞗 = 𝒐𝒑𝒑𝒐𝒔𝒊𝒕𝒆𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆

, 𝑐𝑜𝑠𝞗 = 𝒂𝒅𝒋𝒂𝒄𝒆𝒏𝒕𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆

, 𝑡𝑎𝑛𝞗 = 𝒐𝒑𝒑𝒐𝒔𝒊𝒕𝒆𝒂𝒅𝒋𝒂𝒄𝒆𝒏𝒕

Given the right triangle below, determine the trigonometric ratios.

31. sin 𝐴 = sin𝐵 =

32. cos𝐴 = cos𝐵 =

33. tan𝐴 = tan𝐵 = C. Graphing Trig functions. Using radians, find the amplitude and period of each function. Then graph. 34. 𝑦 = 𝑠𝑖𝑛 𝜃 35. 𝑦 = 3 cos𝜃 36. 𝑦 = !

!tan𝜃

B

A

4140

9C