Saint Basil Academy Mr. Zegestowsky Mathematics · PDF fileSaint Basil Academy Mr. Zegestowsky Mathematics Department Honors Pre-Calculus Summer Review Packet ... Your assignment should

Saint Basil Academy Mr. Zegestowsky Mathematics Department

Honors Pre-Calculus Summer Review Packet Directions: This packet is required if you are a student registered for the Honors Pre-Calculus/Trigonometry, Pre-Calculus/Trigonometry, or Pre-calculus/Introduction to Trigonometry course for the 2015-2016 school year. This packet is due on the first day of class. This packet will be counted as your first five homework assignments. Additionally, failure to submit this packet on the first day in class will result in an immediate penalty of 100 out of 100 points for the citizenship component of your first quarter grade, which constitutes 10% of your quarter grade. Complete your work on separate paper in the order in which it appears in the packet. Show your work and circle/box/highlight your answer. Your assignment should be legible and easy to follow. Attach your work to the review packet. A few high quality resources you may find useful include:

1) http://www.purplemath.com 2) https://www.khanacademy.org

Please take a moment and join the edmodo group as soon as you have read this. We will be using this group for our course of study in the coming school year and it will be your means of communiciating with me if the need arises over the summer as you are completing this assignment.

1) Visit http://www.stbasilacademy.org 2) Under Quick Links select Access Edmodo 3) Use group code nyt5y8 to join Honors Pre-Calculus 2015-2016 or use group

code jhusqe to join Pre-Calculus 2015-2016.

Functions

Operations with Functions Examples:

Find for .

a)

b)

c)

d)

Practice: Perform the following operations for .

1)

2)

3)

4)

5)

6)

Evaluating Functions Examples:

Practice: Evaluate for

7)

8)

9)

10) c −14

"

#$

%

&'

11) k −2( )

f + g( ) x( ) , f − g( ) x( ) , f •g( ) x( ) , fg"

#$

%

&' x( ) f (x) = x2 +3x − 4 and g(x) = 3x − 2

f + g( ) x( )= f (x)+ g(x) = (x2 +3x − 4)+(3x − 2) = x2 + 6x − 6f − g( ) x( )= f (x)− g(x) = (x2 +3x − 4)−(3x − 2) = x2 − 2f •g( ) x( )= f (x)•g(x) = (x2 +3x − 4) (3x − 2) = 3x3 + 7x2 −18x +8fg!

"#

$

%& x( ) = x

2 +3x − 43x − 2

,where x ≠ 23

f (x) = 2x +1 and g(x) = x −3

f + g( ) x( )

g− f( ) x( )

f − g( ) x( )

f •g( ) x( )

fg!

"#

$

%& x( )

gf

!

"#

$

%& x( )

m(x) = 2x2 − x + 2m(2) = 2(2)2 − (2)+ 2m(2) = 8

r(x) = −x3 + x +1r(−4) = −(−4)3 + (−4)+1r(−4) = 61

c(x) = −16x2 − 4x + 2 , h(x) = x4 − 2x , k(x) = 2x3 + 5x2

h(−2)

k(3)

c 12!

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%&

12) h 12!

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Factoring Polynomials Examples: Factor.

a) GCF:

b) Difference of Squares:

c) Product/Sum: d) By Grouping: e) ac-method:

Practice: Factor completely.

13)

14)

15)

16)

17)

18)

19)

20)

21)

22)

4x3y+ 6x2y− 2x = 2x 2x2y+3xy−1( )4x2 − 9 = 2x −3( )(2x +3)x2 + 2x −15= (x + 5)(x −3)x3 − 4x2 + 2x −8 = x2 x − 4( )+ 2(x − 4) = (x2 + 2)(x − 4)

2x2 − 7x +3= 2x2 − x − 6x +3= x(2x −1)−3(2x −1) = (x −3)(2x −1)

5a2b+10ab3

x2 − 25

x2 + 6x

1−16x2

6x3 − 9x2 + 2x −3

5x2 − 20

x2 −8x +15

5x2 − 7x + 2

2x2 − 2x − 24

6x2 − x −1

Solving Quadratic Equations Examples: Solve the following quadratic equations.

a) Solve by factoring:

b) Solve by Quadratic Formula:

Practice: Solve.

23)

24)

25)

26)

27)

28)

x2 + 2x −15= 0x −3( ) x + 5( ) = 0x −3= 0 or x + 5= 0x = 3 or x = −53,−5{ }

2x2 −3x − 9 = 0

x =−(−3) ± (−3)2 − 4(2)(−9)

2(2)=3± 814

=3± 94

x = 124= 3

or

x = − 64= −

32

"

#

$$

%

$$

3,− 32

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x2 − 4x −8 = 0

x =−(− 4) ± (− 4)2 − 4(1)(−8)

2(1)=4± 482

=4± 4 32

x = 2+ 2 3or

x = 2− 2 3

"

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2+ 2 3 , 2− 2 3{ }

x2 + 7x +12 = 0

5x2 =10x

2x2 + x =15

−x2 + 2x +10 = 0

x2 −16 = 0

2x2 −3x −3= 0

Translating Graphs

a) Vertical shift c units upward h(x) = f (x)+ c b) Vertical shift c units downward h(x) = f (x)− c c) Horizontal shift c units to the right h(x) = f (x − c) d) Horizontal shift c units to the left h(x) = f (x + c)

Example:

Practice: 29) f (x) = x2 f (x) = x2 −3 f (x) = x −3( )2 +1

30) f (x) = x3 f (x) = x + 4( )3 f (x) = x3 − 2

Domain and Range of Functions Given a function y = f (x) , the domain is the set of input values x for which the function is defined, which means f (x) results in a number when evaluated. The range is the set of y-values resulting as outputs. Domains and ranges are sets, so use proper notation when identifying both. Domains are typically found algebraically and ranges are typically found graphically. When finding the domain of a function, ask yourself if there are any values of x that cannot be used. The domain is the set of all other values of x. Here are a few simple rules to keep in mind.

a) The domain of all polynomial functions is the set of real numbers. b) Square root functions cannot contain a negative number under the radical sign. So

set the radicand greater than or equal to zero and solve for the variable. That will be your domain.

c) Rational functions cannot have zero denominators. So set the denominator equal to zero and solve for the variable. The domain will be the set of all other values.

Examples: Find the domain for each function.

a) f (x) = 4x − 2

→ Set x − 2 = 0→ x = 2→D = x | x ≠ 2{ }

b) f (x) = 3x2 − 4x

→ Set x2 − 4x = 0→x x − 4( ) = 0→ x = 0, x = 4→D = x | x ≠ 0, 4{ }

c) f (x) = xx2 − 9

→ Set x2 − 9 = 0→ x −3( ) x +3( ) = 0→ x = 3, x = −3→D = x | x ≠ ±3{ }

d) f (x) = x − 9 → Set x − 9 ≥ 0→x ≥ 9→D = x | x ≥ 9{ }

e) f (x) = 4− x → Set 4− x ≥ 0→x ≤ 4→D = x | x ≤ 4{ }

Identifying the domain and range from a graph is relatively easy. For the domain, take the values of x from left to right. For the range, take the values of y from bottom to top.

D = x | − 5< x ≤ 3{ }R = y | − 4 < y ≤ 6{ }

D = x | − 4 < x ≤1{ }R = y | −3≤ y < 5{ }

D = x | − 6 < x < 6{ }R = y | − 2 ≤ y < 5{ }

Practice: Identify the domain for each function.

31) f (x) = 3xx + 2

32) g(x) = x −13x + 2

33) h(x) = 3x − 4

34) f (x) = x2 +3x − 4

35) g(x) = 5− x

36) h(x) = 3x2 +1

37) f (x) = 3x2 − x − 6

Practice: Identify the domain and range for each function.

38)

39)

40)