Advanced Precalculus Summer Assignment The following packet of material contains prerequisite skills and concepts that will be drawn upon from Algebra II as well as a preview of Precalculus. This summer packet is due the first day you return to school in September. All work must be shown for each problem given. This packet will count as a week’s worth of homework assignments. A late penalty will be applied to any work not turned in as indicated. The information contained in this packet will be reviewed at the beginning of the school year and there will be an assessment within the first two weeks of school for which the material in this packet is the basis. This assessment will count as one major test in the Quarter 1 grade.

Name:__________________________________________________ Date:______________________________ Advanced Precalculus 11-5 Summer Assignment

PART 1: Families of Functions Provide a sketch of the function given for each family of functions. Then identify the features of each function.

Families of Functions

Name Sketch of Graph Features Linear Function

General Equation: f x mx b

Example: y x 5 1

Quadratic Function

General Equation: f x ax bx c 2

Example: y x x 22 5 7

Polynomial Function Gen Eq:

...n n

n nf x a x a x a x

1 0

1 0

Example: y x x 3 22 4 1

Direct Variation

General Equation: ; :f x ax D x 0

Example: ,y x x 4 0

Power Function

General Eq: ; :bf x a x D x 0

Example:

, ,y x x or y x x 2 23 0 3 0

b0 b0

Exponential Function

General Equation: xf x a b

Example: .x xy or y 5 2

b 0 1 b 1

Inverse Variation

General Equation: nf x ax

Example: xy 10

n is even

n is odd

Rational Function

General Equation:

P x

Q xf x

where P(x) and Q(x) are polynomials

Example: xxy 12

Families of Functions Practice Questions:

1) Explain why xy 10 can be called a power function.

2) Compare and contrast exponential and power functions. How do they compare algebraically (how do their equations compare)? How do they compare graphically?

3) Compare the graphs of the increasing functions 3xf x and 3g x x . As x , which

function increases at a faster rate? How do the graphs compare as x? For which

values of x is 3 3xx , 3 3xx , 3 3xx .

4) Why is the function 4

3

xr x

x

called a rational function?

5) Sketch a reasonable graph showing how the variables are related and identify the type of function it could be. Explain your answer.

a. The temperature of a cup of coffee and the time since the coffee was poured.

b. The height of a punted football as a function of the number of seconds since it was kicked.

PART 2: Transformations of Functions Activity 1 Objective: Show that you know the effects of various constants on the graph of a function by graphing the transformation and stating the domain and range of the transformed function.

Activity 2 Objective: Explore the graph of the parent function siny x , and transform the graph.

READ BEFORE COMPELTING THIS INVESTIGATION!! Please make sure your graphing calculator is in degree mode. Once you are in degree mode please make sure to set your window to the following settings for x: Xmin = -60 Xmax = 720 Xscl = 30 You should adjust your window for y-values for each problem to see the best graph.

PART 3: Quadratics Review Objective: To solve quadratic equations using various methods, solve quadratic inequalities, write quadratic equations in different forms and model real-world scenarios with quadratics. Activity 1: Solving Quadratics Directions: Solve the equation by factoring, using square roots or using the quadratic formula (if possible). Leave your answers as exact numbers (no rounding).

1) 24 16 0x 2) 23 16 5x x

3) 2 2 5 4x x 4) 2

2 6 36 0x

Activity 2: Completing the Square Directions: Solve the equation by completing the square.

5) 2 2 8 0x x 6) 22 3 5 0x x

Directions: Write the equation in vertex form, 2

f x a x h k

7) 23 12 9f x x x

Activity 3: Solving Quadratic Inequalities Directions: Solve the quadratic inequality algebraically or by graphing.

8) 22 5 12x x 9) 20 2 8x x Directions: Write an equation in one of the following forms.

Vertex Form: 2

f x a x h k

Intercept Form: f x a x p x q

Standard Form: 2f x ax bx c

10) Vertex : 2,1 11)

Passes through the point 3, 2

12) Passes through the points 1,3 , 3, 5 , 0, 2

Activity 4: Modeling with Quadratics Directions: Write and solve a quadratic equation that represented each situation below.

13) Mario’s living room has a length 6 feet wider than the width. The area of the living room is 280square feet. What are the dimensions of his living room?

14) An apple falls from a branch on a tree 30 feet above a man sleeping underneath. When will the apple

strike the man? Recall that the vertical motion model is 2

016h t v t s where t is measured in

seconds, h is measured in feet, 0v is the initial velocity of the object and s is the initial height of the

object.

PART 4 : Trigonometric Functions Activity 1 Objective: Find values of the six trigonometric functions, with or without a calculator.

Practice Problems for Activity 1 Directions: Sketch each angle in standard position. Mark and label the reference angl. Draw a reference triangle and find the value of the six trigonometric functions for each angle. Find one positive and one negative coterminal angle for each

1) 225 2) 7

6

3) 5

2

_____________________________________________________________________________________________________________________________ Directions: Given the following values of sine and cosine, evaluate the indicated trigonometric functions.

sin 48 0.7431

cos 48 0.6691

sin 12 0.2079

cos 12 0.9781

4) cos 132 5) tan 192 6) sin 12

Directions: Use the information below to sketch a triangle in the appropriate quadrant and find the remaining five trigonometric functions.

7) Given that csc 3 and tan 0 , where is an angle in standard position, sketch a triangle and

find the remaining five trigonometric functions of . (6 points)

Activity 2 Objective: To graph a periodic function given the equation or to write an equation given the graph.

Graphing Sine and Cosine Practice Problems