honors precalculus notes packet -...

Honors Precalculus

Notes Packet 2013-2014

Academic Magnet High School

Contents Unit 1: Inequalities, Equations, and Graphs ................................................................................................. 1

Unit 2: Functions and Graphs ..................................................................................................................... 16

Unit 3: New Functions from Old ................................................................................................................. 37

Unit 4: Polynomial and Rational Functions ................................................................................................. 50

Unit 5: Graphs of Functions – Revisited ...................................................................................................... 71

Unit 6: Trigonometry – Part1 ...................................................................................................................... 80

Unit 7: Trigonometry – Part2 ...................................................................................................................... 94

Unit 8: Inverse Trigonometric Functions .................................................................................................. 104

Unit 9: Sequences and Series .................................................................................................................... 113

Unit 10: Exponential and Logarithmic Functions ...................................................................................... 122

1 AMHS Precalculus - Unit 1

Unit 1: Inequalities, Equations, and Graphs

Interval Notation

Interval notation is a convenient and compact way to express a set of numbers on the real number line.

Graphic Representation

___________________________

___________________________

___________________________

___________________________

___________________________

Inequality Notation

2 3x

1 4x

1 2x

2x

1x

Interval notation

Inequality Properties

1. If a b , then a c b c

2. If a b and 0c , then ac bc

3. If a b and 0c , then ac bc

Ex. 1 Solve each inequality (note that the degree is 1) and write the solution using interval notation:

a) 3 5 12x

b) 9 2 10 5x c) 7 2

3 43

x


Ex. 2 Solve each inequality and write the solution using inequality notation.

a) 0 2 23

x

b) 0 22

x c) 0 2 2x

Polynomial Inequalities with degree two or more and Rational Inequalities

Solve 2 4 7 4x x by making a sign chart. Write your answer using interval notation.

1. Set one side of the inequality equal to zero.

2. Temporarily convert the inequality to an equation.

3. Solve the equation for x . If the equation is a rational inequality, also determine the values of x

where the expression is undefined (where the denominator equals zero). These are the partition

values.

4. Plot these points on a number line, dividing the number line into intervals.

5. Choose a convenient test point in each interval. Only one test point per interval is needed.

6. Evaluate the polynomial at these test points and note whether they are positive or negative.

7. If the inequality in step 1 reads 0 , select the intervals where the test points are positive. If the

inequality in step 1 reads 0 , select the intervals where the test points are negative.


Ex. 3 Solve each inequality. Show the sign chart. Draw the solution on the number line and express the

answer using interval notation.

a) 2( 4)( 3) 0x x x

b) 2 3 4 0x x

c) 2

30

4

x

x d)

3 2

4 1x x


Absolute Value

0

0

x if xx

x if x

The absolute value of a real number x is the distance on the number line that x is from 0.

Absolute value equations

Ex. 4 Solve the equation (check your answers for extraneous solutions):

a) 2 1

43

x

x b) 2 3 1x

Absolute value inequalities

1. if x a , then a x a __________________________________

2. if 0x a , then x a or x a __________________________________

Ex. 5 Solve the inequality. Express your answers in interval notation and graph the solution:

a) 4 1 .01x b) 2 1 5x


c)

Equations and Graphs

Lines

The equation y mx b is a linear equation where m and b are constants. This is called Slope-

Intercept form where m is the slope and b is the y-intercept.

In general,

0m 0m 0m m is undefined


The slope of a Line

Point-Slope equation of a line:

Ex. 1 Find the point-slope equation of a line passing through the points (-1, -2) and (2,5).

Ex. 2 Write the equation of a line passing through the points (4,7) and (0,3).


Parallel and Perpendicular Lines

Two non-vertical lines are parallel iff they have the same slope.

Two lines with non-zero slopes 1m and 2m are perpendicular iff 1 2 1m m .

Ex. 3 Find the equation of the line passing through the point (-3,2) that is parallel to 5 2 3x y .

Ex. 4 Find the equation of the line passing through (-4,3) which is perpendicular to the line passing

through (-3,2) and (1,4).

Ex. 5 A new car costs $29,000. Its useful lifetime is approximately 12 years, at which time it will be worth

an estimated $2000.00.

a) Find the linear equation that expresses the value of the car in terms of time.

b) How much will the car be worth after 6.5 years?


Ex. 6 The manager of a furniture factory finds that it costs $2220 to manufacture 100 chairs and $4800

to manufacture 300 chairs.

a) Assuming that the relationship between cost and the number of chairs produced is linear, find

an equation that expresses the cost of the chairs in terms of the number of chairs produced.

b) Using this equation, find the factory’s fixed cost (i.e. the cost incurred when the number of

chairs produced is 0).

Ex. 7 Find the slope-intercept equation of the line that has an x-intercept of 3 and a y-intercept of 4.


Circles

Recall the distance formula 2 2

2 1 2 1( ) ( )d x x y y

The Standard form for the equation of a circle is:

Ex.1 Write the equation of a circle with center (-1,2) and radius 3. Sketch this circle.

Ex.2 Write the equation of a circle with center at the origin and radius 1.

Ex.3 Find the equation of the circle with center (-4,1) that is tangent to the line x = -1.


Ex. 4 Find the equation of the circle with center (4,3) and passing through the point (1,4).

Ex. 5 Express the following equations of a circle in standard form. Identify the center and radius:

a) 2 2 4 6 3x y x y

b) 2 22 4 4x x y y


The intercepts of a graph

The x -coordinates of the x - intercepts of the graph of an equation can be found by setting 0y and

solving for x .

The y -coordinates of the y - intercepts of the graph of an equation can be found by setting 0x and

solving for y .

Ex. 1 Find the x and y intercepts of the line and sketch its graph: 2 1x y

Ex. 2 Find the x and y intercepts of the circle and sketch its graph: 2 2 9x y

Ex. 3 Find the intercepts of the graphs of the equations.

a) 2 2 9x y

b) 22 5 12y x x


Symmetry

In general :

A graph is symmetric with respect to the y axis if whenever ( , )x y is on a graph ( , )x y is also

a point on the graph.

A graph is symmetric with respect to the x axis if whenever ( , )x y is on a graph ( , )x y is also


A graph is symmetric with respect to the origin if whenever ( , )x y is on a graph ( , )x y is also


Tests for Symmetry:

The graph of an equation is symmetric with respect to:

a) the y axis if replacing x by x results in an equivalent equation.

b) the x axis if replacing y by y results in an equivalent equation.

c) the origin if replacing x and y by x and y results in an equivalent equation.

Ex. 1 Show that the equation 2 3y x has y axis symmetry.


Ex. 2 Show that the equation 2 10x y has x axis symmetry.

Ex. 3 Show that the equation 2 2 9x y has symmetry with respect to the origin.

Ex. 4 Find any intercepts of the graph of the given equation. Determine whether the graph of the

equation possesses symmetry with respect to the x axis, y axis, or origin.

a) 2x y

b) 2 4y x

c) 2 2 2y x x

d) 9y x


Algebra and Limits

Difference of two squares: 2 2 ( )( )a b a b a b

Difference of two cubes: 3 3 2 2( )( )a b a b a ab b

Sum of two cubes: 3 3 2 2( )( )a b a b a ab b

Binomial Expansion 2 2 22 : ( ) 2n a b a ab b

Binomial Expansion 3 3 2 2 33: ( ) 3 3n a b a a b ab b

Limits

Ex. 1 Estimate 22

2lim

4x

x

x numerically by completing the following chart:

x y x y

1.9 2.1

1.99 2.01

1.999 2.001

Conclusion: 22

2lim

4x

x

x=

Properties of Limits

If a and c are real numbers, then lim , lim , lim n n

x a x a x ac c x a x a

Ex. 2 Find the limit:

a) 3

2lim( 4)x

x x b) 1

lim(2 7)x

x


Ex. 3 Find the given limit by simplifying the expression

a) 2

22

6lim

5 6x

x x

x x

b) 2

32

4lim

8x

x

x

c) 1

2 3lim

1x

x

x

d) 2

2

7 10lim

2x

x x

x

e) 22

2lim

5 3x

x

x

f) 0

1 1

8 8limx

x

x


Unit 2: Functions and Graphs

Functions

A function is a rule that assigns each element in the domain to exactly one element in the range.

The domain is the set of all possible inputs for the function. On a graph these are the values of the

independent variable (most commonly known as the x values).

The range is the set of all possible outputs for the function. On a graph these are the values of the

dependent variable (most commonly known as the y values).

We use the notation ( )f x to represent the value (again, in most cases, a y - value) of a function at the

given independent value of x . For any value of x , ( , ( ))x f x is a point on the graph of the function

( ).f x

Ex. 1 Given 2( )f x x , graph the function and determine the domain and range. Use interval notation

to express the domain and range.


Ex. 2 Given ( )f x x , graph the function and determine the domain and range. Use interval notation

to express the domain and range.

Ex. 3 For the function 2( ) 2 4f x x x , find and simplify:

a) a) ( 3)f b) b) ( )f x h

Ex. 4 For 2 , 0

( )2 1, 0

x xf x

x xfind:

a) (1)f b) ( 1)f

c) ( 2)f d) (3)f


Ex. 5 The graph of the function f is given:

a) Determine the values: ( 2)f (0)f

(2)f (4)f

b) Determine the domain:

c) Determine the range:

Ex. 6 The graph of the function f is given:

a) ( 3)f (0)f (4)f

b) For what numbers x is ( ) 0?f x

c) What is the domain of f ?

d) What is the range of f ?

e) What is (are) the x -intercept(s)?

f) What is the y - intercept?

g) For what numbers x is ( ) 0?f x

h) For what numbers x is ( ) 0?f x


Vertical Line Test for a Function: An equation is a function iff every vertical line intersects the graph of

the equation at most once.

Ex. 7 Determine which of the curves are graphs of functions:

a)

b)

c)

Domain (revisited)

Rule for functions containing even roots (square roots, 4th roots, etc):

Ex. 1 Determine the domain and range of ( ) 4 3f x x

Ex. 2 Determine the domain of 2( ) 2 15f t t t


Rule for functions containing fractional expressions:

Ex. 3 Determine the domain of 2

5( )

3 4

xh x

x x


1( )

2 15g x

x x


( )2

xh x

x


Intercepts (revisited)

The y -intercept of the graph of a function is (0, (0))f .

The x - intercept(s) of the graph of a function ( )f x is/are the solution(s) to the equation ( ) 0.f x

These x - values are called the zeros of the function ( )f x .

Ex. 1 Find the zeros of ( ) (3 1)( 9)f x x x x

Ex. 2 Find the zeros of 2( ) 5 6f x x x

Ex. 3 Find the zeros of 4( ) 1f x x

Ex. 4 Find the x - and y - intercepts (if any) of the graph of the function 1

( ) 42

f x x


Ex. 5 Find the x - and y - intercepts (if any) of the graph of the function 2( ) 4( 2) 1f x x

Ex. 6 Find the x - and y - intercepts (if any) of the graph of the function 2

2

4( )

16

xf x

x

Ex. 7 Find the x - and y - intercepts (if any) of the graph of the function 23( ) 4

2f x x


Transformations – Horizontal and Vertical shifts

Suppose ( )y f x is a function and c is a positive constant. Then the graph of

1. ( )y f x c is the graph of f shifted vertically up c units.

2. ( )y f x c is the graph of f shifted vertically down c units.

3. ( )y f x c is the graph of f shifted horizontally to the left c units.

4. ( )y f x c is the graph of f shifted horizontally to the right c units.

Ex. 1 Consider the graph of a function ( )y f x shown on the coordinates. Perform the following

transformations.

( ) 3y f x ( ) 2y f x

( 1)y f x

( 3)y f x


Suppose ( )y f x is a function. Then the graph of

1. ( )y f x is the graph of f reflected over the x -axis.

2. ( )y f x is the graph of f reflected over the y -axis.

Ex. 2 Consider the graph of a function ( )y f x . Sketch ( 2) 3y f x

Common (Parent) Functions

( )f x x

2( )f x x


( )f x x

3( )f x x

3( )f x x

1( )f x

x

( )f x x or x


Combining common functions with transformations

Sketch the graphs of the following functions. Determine the domain and range and any intercepts.

Ex. 1 ( ) 2 1f x x

Ex.2 ( ) 1 2f x x

Ex. 3 3( ) ( 2) 1f x x

Ex. 4 ( ) 1 3f x x


Symmetry (revisited)

Tests for Symmetry

The graph of a function f is symmetric with respect to:

1. the y -axis if ( ) ( )f x f x for every x in the domain of the ( )f x .

2. The origin if ( ) ( )f x f x for every x in the domain of the ( )f x .

If the graph of a function is symmetric with respect to the y -axis, we say that f is an even function.

If the graph of a function is symmetric with respect to the origin, we say that f is an odd function.

In examples 1-3, determine whether the given function ( )y f x is even, odd or neither. Do not graph.

Ex. 1 5 3( )f x x x x

Ex.2 ( ) 23f x x

Ex. 3 2( ) 2f x x x


Transformations – Vertical Stretches and Compressions

Suppose ( )y f x is a function and c a positive constant. The graph of ( )y cf x is the graph of f

1. Vertically stretched by a factor of c if 1c

2. Vertically compressed by a factor of c if 0 1c

Ex.1 Given the graph of ( )y f x

a) Sketch 2 ( )y f x b)

1( )

2y f x

Ex. 2 Sketch the graph of the following functions. Include any intercepts.

( ) 1f x x

( ) 3( 1)f x x


Quadratic Functions

A quadratic function ( )y f x is a function of the form 2( )f x ax bx c where 0a , b and c are

constants.

The graph of any quadratic function is called a parabola.

The graph opens upward if 0a and downward if 0a .

The domain of a quadratic function is the set of real numbers ( , ) .

A quadratic function has a vertex (which serves as the minimum or maximum of the function depending

on the value of a ), a line of symmetry, and may have zero, one or two x - intercepts.

Ex. 1 Sketch the graph of 2( ) ( 1) 3f x x . Determine any intercepts.


The standard form of a quadratic function is 2( ) ( )f x a x h k where ( , )h k is the vertex of the

parabola and x h is the line of symmetry.

Ex. 2 Rewrite the quadratic function 2( ) 2 3f x x x in standard form by completing the square.

Determine any intercepts, the vertex, the line of symmetry and sketch the graph.

Ex. 3 Rewrite the quadratic function 2( ) 4 12 9f x x x in standard form by completing the square.

Determine any intercepts, the vertex, the line of symmetry and sketch the graph.


Ex. 4 Complete the square to find all the solutions to the equation 2 0ax bx c

The vertex of any parabola of the form 2( )f x ax bx c is ( , ( ))2 2

b bf

a a.

Ex. 5 Find the vertex of the quadratics from examples 2 and 3 directly by using ( , ( ))2 2

b bf

a a.

Ex. 6 Find the vertex from example 2 by using the x - intercepts and the line of symmetry.


Ex.7 Find the intercepts and vertex of the function 21( ) 1

2f x x x

Ex. 8 Find the maximum or the minimum of the function.

1. 2( ) 3 8 1f x x x

2. 2( ) 2 6 3f x x x

Ex.9 Determine the quadratic function whose graph is given.


Freely Falling Object - Suppose an object, such as a ball, is either thrown straight upward or downward

with an initial velocity 0v or simply dropped ( 0 0v ) from an initial height 0s . Its height, ( )s t as a

function of time t can be described by the quadratic function 2

0 0

1( )

2s t gt v t s

Gravity on earth is 232 / secft or 29.8 / secm .

Also, the velocity of the object while it is in the air is 0( )v t gt v

Ex. 10 An arrow is shot vertically upward with an initial velocity of 64 / secft from a point 6 feet above

the ground.

1. Find the height ( )s t and the velocity ( )v t of the arrow at time 0t .

2. What is the maximum height attained by the arrow? What is the velocity of the arrow at the

time it attains its maximum height?

3. At what time does the arrow fall back to the 6 foot level? What is its velocity at this time?

Ex. 11 The height above the ground of a toy rocket launched upward from the top of a building is given

by 2( ) 16 96 256s t t t .

1. What is the height of the building?

2. What is the maximum height attained by the rocket?

3. Find the time when the rocket strikes the ground. What is the velocity at this time?


Horizontal Stretches and Compressions

Suppose ( )y f x is a function and c a positive constant. The graph of ( )y f cx is the graph of f

1. Horizontally compressed by a factor of 1

cif 1c

2. Horizontally stretched by a factor of 1

cif 0 1c

Ex.1 Given the graph of ( )y f x

c) Sketch (2 )y f x d)

1( )2

y f x

Ex.2 Consider the function 2( ) 4f x x

a) On the same axis, sketch ( ), 2f x f x and 1

( )2

f x . Identify any intercepts of each function.


b) On the same axis, sketch ( ), 2f x f x and 1

( )2

f x . Identify any intercepts of each function.

List the transformations on ( )f x x required to sketch ( ) 2 1 2f x x


Silly String Activity

Objective: The use a quadratic function to model the path of silly string.

Materials: Can of silly string, tape measure, stopwatch, clear overhead transparency, TI84

Personnel: Timekeeper, Silly-String operator, assistant

Calculate the initial velocity 0v of the silly string as it exits the can.

1. Hold the can of silly string 1 foot above the ground. Have the timekeeper start the stopwatch

and say “go”. At this time, shoot a short burst of silly string towards the ceiling. Have the class

keep a casual eye on the maximum height the silly string achieves. When the silly string hits the

floor, have the timekeeper stop the stopwatch and record the elapsed time.

2. Measure the maximum height of the silly string observed by the class. Use the position equation

2

0 0

1( )

2s t gt v t s

with g = 232 / secft

to calculate 0v . ( 0s = 1, get t from the timekeeper. This represents the time it took for the silly

string to reach the ground, i.e. ( )s t =0)

Now that we know 0,g v and 0s we can set up a position equation to model the height of the silly string

as a function of time. Use this equation to determine the maximum height (the vertex holds this info) of

the silly string. How does this compare to the actual height observed by the class. What factors might

have caused it to be different?

Now we are going to get the assistant to lean over the can of silly string (with the clear overhead

transparency protecting the face) in its original position 1 foot above the ground and see if the assistant

can move fast enough to avoid getting silly string in the face. Calculate the time it would take for the

silly string to reach the assistant’s face (set ( )s t = the height of the assistant’s face and solve for t )

Once the reaction time for the assistant has been calculated and discussed, see if the assistant can

actually react that quickly, i.e. avoid silly string in the face.

To date, it has never been done. Enjoy!


Unit 3: New Functions from Old

Piecewise – Defined Functions

A function f may involve two or more functions, with each function defined on different parts of the

domain of f . A function defined in this manner is called a piecewise-defined function.

Ex.1 Sketch the graph of the given function and find the following:

0( )

0

x if xf x x

x if x

a) ( 1)f

b) (2)f

c) Domain:

d) Range:

Ex. 1b Express ( ) 3f x x as a piecewise function:

Ex.2 Sketch the graph of the given function and find the following:

2

3 1 1( )

1 0

x if xf x

x if x

a) ( 1)f

b) (2)f

c) Domain:

d) Range:


Ex.3 Graph the following

a)

1 0

( ) 0 0

1 0

if x

f x if x

x if x

b)

1( )

1

xf x

x

Hint: write this as a piecewise function

Domain:

Domain:

Range: Range:


Graphing the Absolute Value of a Function

Sketch the graph of the given functions. Include any intercepts.

Ex.1 Ex.2

0( )

0

x if xf x x

x if x

2( ) ( 2) 4f x x

Ex.3 Ex.4 2( ) 4 3f x x x ( ) 3 1f x x


Compositions of Functions

The composition of the function f with the function g , denoted f g is defined by

( )( ) ( ( ))f g x f g x . The domain of f g consists of those x values in the domain of g for which

( )g x is in the domain of f .

Ex. 1 ( ) 2f x x and 2( ) 1g x x . Find the following:

a) ( )( )f g x

b) Find the domain of ( )( )f g x

c) ( )( )g f x

d) Find the domain of ( )( )g f x

e) ( )(2)f g

f) ( )(4)g f

g) ( )(1)g f

Ex.2 Write the function 2( ) 3f x x as the composition of two functions

Ex.3 Write the function 2

3( )

4 1f x

x xas the composition of three functions.


Ex. 4 Given 2( ) ( 4) 4F x x x ¸find functions f and g such that ( ) ( )( )F x f g x .

Ex. 5 A metal sphere is heated so that t seconds after the heat had been applied, the radius ( )r t is given

by ( ) 3 .001r t t cm. Express the Volume of the sphere as a function of t .

Ex. 6 ( )f x x and 2( ) , ( 0)g x x x . Find the following:

a) ( )( )f g x

b) ( )( )g f x

c) (3)g

d) (4)g

e) (9)f

f) (16)f


Inverse Functions

Suppose that f is a one-to-one function with domain X and range Y . The inverse function for the

function f is the function denoted 1f with domain Y and range X and defined for all values x X by

1( ( ))f f x x and 1( ( ))f f x x .

Ex. 1 Prove that ( ) 2f x x and 2( ) 2g x x ( 0)x are inverse functions using composition.

Ex. 2 Find the inverse of 3

( )( 4)

xf x

xand check using composition. Find the domain and range of

( )f x and 1( )f x .

Steps for Finding the Inverse of a Function:

1. Set ( )y f x

2. Change x y and y x

3. Solve for y

4. Set 1( )y f x


The graph of 1( )f x is a reflection of the graph of ( )f x about the line y x .

One-to-One Functions

A function is one-to-one iff each number in the range of f is associated with exactly one number in its

domain. In other words, 1 2( ) ( )f x f x implies 1 2x x .

Horizontal Line Test for One-to-One Functions

A function is one-to-one precisely when every horizontal line intersects its graph at most once.

Ex. 3 Determine whether the given function is one-to-one

a) 3( ) 2f x x b) 2( ) 2f x x x


Ex.4 Given ( ) 2 3f x x

Domain of ( )f x :

Domain of 1( )f x :

Range of ( )f x : Range of 1( )f x :

Find 1( )f x and check using composition.

Sketch the graph of 1( )f x and ( )f x on the same axis.


Translating Words into Functions

In calculus there will be several instances where you will be expected to translate the words that

describe a problem into mathematical symbols and then set up or construct an equation or a function.

In this section, we will focus on problems that involve functions. We begin with a verbal description

about the product of two numbers.

Ex.1 The sum of two nonnegative numbers is 15. Express the product of one and the square of the other

as a function of one of the numbers.

Ex.2 A rectangle has an area of 400 2in . Express the perimeter of the rectangle as a function of the

length of one of its sides.

Ex.3 Express the area of a circle as a function of its diameter d .


Ex. 4 An open box is made from a rectangular piece of cardboard that measures 30cm by 40cm by

cutting a square of length x from each corner and bending up the sides. Express the volume of the box

as a function of x .

Ex. 5 Express the area of the rectangle as a function of x . The equation of the line is 2 4x y .The

lower left-hand corner is on the origin and upper right-hand corner of the rectangle with coordinate

( , )x y is on the line.

Ex. 6 Express the area of an equilateral triangle as a function of the length s of one of its sides.


The Tangent Line Problem

Find a tangent line to the graph of a function f .

tan0

( ) ( )limx

f a x f am

x

Ex.1 Find the slope of the tangent line to the graph of 2( ) 2f x x at 1x .




The DERIVATIVE of a function ( )y f x is the function 'f defined by:

0

( ) ( )'( ) lim

x

f x x f xf x

x

Ex.4 Find the derivative of 2( ) 2f x x .

Ex.5 Find the derivative of 2( ) 2 6 3f x x x and use it to find the slope and then the equation of

the tangent line at 2x .


Ex. 6 Find the slope of the tangent line to the graph of 2

( )f xx

at 1x .

Ex.7 Find the derivative of 2

( )f xx

and use it to find the slope and then the equation of the tangent

line at 2x .

Ex. 8 Find the derivative of ( ) 2f x x .


Unit 4: Polynomial and Rational Functions

Polynomial Functions

A polynomial function ( )y p x is a function of the form

1 2 2

1 2 2 1 0( ) ...n n n

n n np x a x a x a x a x a x a

where 1 2 1 0, ,..., , ,n na a a a a are real constants and are called the coefficients of ( )p x .

n is the degree of ( )p x and is a positive integer.

na is called the leading coefficient and 0a is the constant term of the polynomial.

The domain of any polynomial is all real numbers.

Ex. 1 Determine the degree, the leading coefficient and the constant term of the polynomial.

a) 4 3( ) 5 7 3 7f x x x x b) 3 2( ) 13 5 4g x x x x

End Behavior of a Polynomial

There are four scenarios:

1) Sketch 2 4( ) , ( )p x x p x x ( n is even,

0na )

2) Sketch 2 4( ) , ( )p x x p x x ( n is

even, 0na )

As , ( )x p x As , ( )x p x

As , ( )x p x As , ( )x p x


3) Sketch 3 5( ) , ( )p x x p x x ( n is odd,

0na )

4) Sketch 3 5( ) , ( )p x x p x x ( n is odd,

0na )

As , ( )x p x , ( )x p x

As , ( )x p x , ( )x p x

As x and x , the graph of the polynomial 1 2 2

1 2 2 1 0( ) ...n n n

n n np x a x a x a x a x a x a resembles the graph of n

ny a x .

Ex. 2 Use the zeros and the end behavior of the polynomial to sketch an approximation of the graph of

the function.

a) 3( ) 9f x x x b) 4 2( ) 5 4g x x x


c) 5( )f x x x

Repeated Zeros

If a polynomial ( )f x has a factor of the form ( )kx c , where 1k , then x c is a repeated zero of

multiplicity k .

If k is even, the graph of ( )f x flattens and just touches the x -axis at .x c

If k is odd, the graph of ( )f x flattens and crosses the x -axis at .x c

Ex. 4: Sketch the given graphs

a) 4 3 2( ) 3 2f x x x x b) 3( ) ( 1) ( 2)( 3)g x x x x


Ex. 5: The cubic polynomial ( )p x has a zero of multiplicity two at 1x , a zero of multiplicity one at

2x , and ( 1) 2p . Determine ( )p x and sketch the graph.

Ex. 6: An open box is to be made from a rectangular piece of cardboard that is 12 by 6 feet by cutting

out squares of side length x feet from each corner and folding up the sides.

a) Express the volume of the box ( )v x as a function of the size x cut out at each corner.

b) Use your calculator to approximate the value of x which will maximize the volume of the box.

Ex. 7: The product of two non-negative numbers is 60. What is the minimum sum of the two numbers?


The Intermediate Value Theorem

Suppose that f is continuous on the closed interval [ , ]a b and let N be any number between ( )f a and

( )f b , where ( ) ( )f a f b . Then there exists a number c in ( , )a b such that ( )f c N .

Ex. 1: Verify that the Intermediate Value Theorem applies to the indicated interval and find the value of

c guaranteed by the theorem.

2( ) 1f x x x , [0,5], ( ) 11f c

Ex. 2: Show that there is a root of the equation 3 2 1 0x x in the interval (0,1).


The Division Algorithm

Let ( )f x and ( ) 0d x be polynomials where the degree of ( )f x is greater than or equal to the degree

of ( )d x . Then there exists unique polynomials ( )q x and ( )r x such that

( ) ( )( )

( ) ( )

f x r xq x

d x d x or ( ) ( ) ( ) ( )f x d x q x r x .

where ( )r x has a degree less than the degree of ( )d x .

Ex. 1: Divide the given polynomials.

a) 3 26 19 16 4

2

x x x

x

b) 3 1

1

x

x

c) 3 2

2

3 2 6

1

x x x

x


Remainder Theorem

If a polynomial ( )f x is divided by a linear polynomial x c , then the remainder r is the value of ( )f x

at x c . In other words , ( )f c r

Ex. 2: Use the Remainder Theorem to find r when 3 2( ) 4 4f x x x is divided by 2x .

Ex. 3: Use the Remainder Theorem to find ( )f c for 4 2( ) 3 5 27f x x x when 1

2c

Synthetic Division

Synthetic division is a shorthand method of dividing a polynomial ( )p x by a linear polynomial x c . It

uses only the coefficients of ( )p x and must include all 0 coefficients of ( )p x as well.

Ex. 4: Use synthetic division to find the quotient and remainder when

a) 3( ) 1f x x is divided by 1x

b) 4 2( ) 14 5 9f x x x x is divided by 4x

c) 4 3 28 30 23 8 3x x x x is divided by 1

4x


Ex. 5: Use synthetic division and the Remainder Theorem to find ( )f c for

6 5 4 3 2( ) 3 4 8 6 9f x x x x x x when 2c .

Ex. 6: Use synthetic division and the Remainder Theorem to find ( )f c for

3 2( ) 7 13 15f x x x x when 5c .

The Factor Theorem

A number c is a zero of a polynomial ( )p x ( ( ) 0p c ) if and only if ( )x c is a factor of ( )p x .

Ex. 1: Determine whether

a) 1x is a factor of 4 2( ) 5 6 1f x x x x

b) 2x is a factor of 3 23 4x x


Fundamental Theorem of Algebra

A polynomial function ( )p x of degree 0n has at least one zero.

In fact, every polynomial function ( )p x of degree 0n has at exactly n zeros.

Complete Factorization Theorem

Let 1 2, ,... nc c c be the n (not necessary distinct) zeros of the polynomial function of degree 0n :

1 2 2

1 2 2 1 0( ) ...n n n

n n np x a x a x a x a x a x a .

Then ( )p x can be written as the product of n linear factors

1 2( ) ( )( ) ( ).n np x a x c x c x c

Ex.1: Give the complete factorization of the given polynomial ( )p x with given information:

a) 3 2( ) 2 9 6 1p x x x x ; 1

2x is a zero.

b) 4 3 2( ) 4 8 61 2 15p x x x x x ; 3, 5x x are both zeros.


c) 3 2( ) 6 16 48p x x x x ; ( 2)x is a factor.

d) 4 3 2( ) 3 7 5p x x x x x ; (3 1)x x is a factor.

Ex. 2: Find a polynomial function ( )f x of degree three, with zeros 1,-4, 5 such that the graph possesses

the y - intercept (0,5) .


The Rational Zero Test

Suppose p

qis a rational zero of 1 2 2

1 2 2 1 0( ) ...n n n

n n nf x a x a x a x a x a x a ,

where 0 1, ......, na a a are integers and 0na . Then p divides 0a and q divides na .

The Rational Zero Test provides a list of possible rational zeros.

Ex. 1: Find all the rational zeros of ( )f x then factor the polynomial completely.

a) 4 3 2( ) 3 10 3 8 2.f x x x x x

b) 4 3 2( ) 3 3 2f x x x x x


Complex Roots of Polynomials

Consider factoring the function:

3( ) 1f x x

The Square Root of -1

We define

1i so that

2 1.i

Complex Numbers

A complex number is a number of the form a bi where a and b are real numbers. The number a is

called the real part and the number b is called the imaginary part.

Complex Arithmetic

Ex. 1

a) (2 3 ) (6 )i i

b) (2 3 )(4 )i i

c) (3 6 )(3 6 )i i d) (4 5 )(4 5 )i i

Complex Conjugates

The complex conjugate for a complex number z a bi is z a bi .

In general, ( )( )a bi a bi


Ex. 2: Simplify.

a) (2 3 )

(1 6 )

i

i

(2 )

(1 7 )

i

i

Ex. 3: Simplify.

a) 4 b) 8

Ex. 4: Determine all solutions to the equation 2 4 13 0x x

Ex. 5: Completely factor 3( ) 1.f x x


Ex. 6: Find the complete factorization of 4 3 2( ) 12 47 62 26f x x x x x given that 1 is a zero of

multiplicity two.

Conjugate Pairs of Zeros of Real Polynomials

If the complex number z a bi is a zero of some polynomial ( )p x with real coefficients, then its

conjugate z a bi is also a zero of ( )p x .

Ex. 7: Find a 3rd degree polynomial ( )g x with real coefficients and a leading coefficient of 1 with zeros 1

and 1 i .

Ex. 8: 1 2i is a zero of 4 3 2( ) 2 4 18 45.f x x x x x Find all other zeros and then give the

complete factorization of ( )f x .


Rational Functions

A rational function ( )y f x is a function of the form ( )

( ) ,( )

p xf x

q xwhere p and q are polynomial

functions.

Ex. 1: Recall the parent function1

( )f xx

. Use transformations to sketch 2

( )1

g xx

Asymptotes of Rational Functions

The line x a is a vertical asymptote of the graph of ( )f x if ( )f x or ( )f x as x a

(from the right) or x a (from the left).

Vertical Asymptotes

The graph of ( )

( )( )

p xf x

q xhas vertical asymptotes at the zeros of ( )q x after all of the common factors

of ( )p x and ( )q x have been canceled out; the values of x where ( ) 0q x and ( ) 0p x .

Holes

The graph of ( )

( )( )

p xf x

q xhas a hole at the values of x where ( ) 0q x and ( ) 0p x .


Horizontal Asymptotes

The line y b is a horizontal asymptote of the graph of ( )f x if ( )f x bwhen x or x .

In particular, with a rational function 1

1 1 0

1

1 1 0

...( )( )

( ) ...

n n

n n

m m

m m

a x a x a x ap xf x

q x b x b x b x b

There are three cases:

1. If n m , then 0y is the horizontal asymptote.

Ex: 3

3( )

13 7

xf x

x x

2. If n m , then n

m

ay

bis the horizontal asymptote.

Ex: 3

3 2

3 6( )

4 3

x xf x

x x

3. If n m , then there is no horizontal asymptote.

Ex: 4 3 2

2

3 2 5 1( )

3 4

x x xf x

x x

Slant Asymptote

If the degree of numerator is exactly one more than the degree of the denominator, the graph of ( )f x

has a slant asymptote of the form y mx b . The slant asymptote is the linear quotient found by

dividing ( )p x by ( )q x and essentially disregarding the remainder.

Ex: 3

2

8 12( )

1

x xf x

x


Ex. 2: Find all asymptotes and intercepts and sketch the graphs of the given rational functions:

a) 2

( )1

f xx

Domain:

Range:

Equation(s) of vertical asymptotes:

Equation(s) of horizontal asymptotes:

Equation of slant asymptote:

x - intercepts:

y - intercept:

b) 3 2

( )2 4

xf x

x

Domain:

Range:




x - intercepts:

y - intercept:


c) 3

( )( 2)( 5)

xf x

x x

Domain:

Range:




x - intercepts:

y - intercept:

d) 2

( )1

x xf x

x

Domain: Equation(s) of vertical asymptotes:



x - intercepts:

y - intercept:


e) 2

1( )

2

xf x

x x

Domain:

Range:




x - intercepts:

y - intercept:

f) (3 1)( 2)

( )( 2)( 1)

x xf x

x x

Domain:

Range:




x - intercepts:

y - intercept:

Ex. 3: Sketch the graph of a rational function that satisfies all of the following conditions:

( )f x as 1x and ( )f x as 1x

( )f x as 2x and ( )f x as 2x

( )f x has a horizontal asymptote 0y

( )f x has no x -intercepts

Has a local maximum at ( 1, 2)


Honors Precalculus – Academic Magnet High School Name_____________________

Mandelbrot Set Activity

using Fractint fractal generator

STEP 1 - CREATE, SAVE, and PRINT an inspirational, visually pleasing area of the Mandelbrot set.

Important Menu Items:

VIEW- Image Settings, Zoom In/ Out box, Coordinate Box

FRACTALS- Fractal Formula, Basic Options, Fractal Parameters

COLORS- Load Color- Map

FILE- Save As

1) Start Fractint by clicking on the desktop icon.

Fractint always starts with the Mandelbrot set, but in case things get weird, ALWAYS

make sure “mandel” is selected in the Fractal-Fractal Formula menu item.

Use the Image Settings box to set the size of the picture (800 x 600 should work fine).

2) Use the Zoom In/Out feature along with the Colors-Load Color Map to create a variation of

the Mandelbrot set.

If the color palettes do not load, double click on the box that is labeled Pallette Files

(*.Map)

If you zoom in a few times you lose detail, you can increase the iterations in the

Fractals-Basic Options Box- Remember that the more iterations the computer has to

perform, the longer it will take

3) Use the Fractals-Fractal Params …window to record the x and y mins and maxs of the

viewing rectangle on the imaginary plane.

4) Using the Coordinates box, point your arrow to a point you think is in the Mandelbrot set and

record the x and y values.

5) Repeat #4 for a point you think is NOT in the set.

6) SAVE the fractal. Write down the coordinates (x and y mins and maxs) and number of

iterations of your current position in the Mandelbrot set.

7) Print your fractal.

STEP 2 - Create a typed text document (1 page or so) including, but not limited to:

The NAME of your group’s fractal and the name of everyone in your group

A short story about your creation (what it makes you think of, color, choice, etc.)


STEP 3 – Typed:

1) List the x and y mins and maxs for your viewing rectangle from Step 1

2) Recall the coordinates of the point you thought was in the Mandelbrot set from Step 1.

Let x = a and y = b for the complex number a + bi

Let this number a + bi = c iterate this value 100 or more times using the Mandelbrot

sequence:

x0 = c

x1 = x02 + c

x2 = x12 + c

Etc…

You will be using decimals and your calculator. Unlike the fractals, these calculations will

not be pretty. Let your TI-84 do the work for you (i is above the decimal point).

3) Record the last 20 iterations for analysis. Remember that you may need to scroll the TI- 84

to the right to get the entire number

4) Were your predictions right about this point? Do you need more information to determine if

it is in the set?

5) Repeat for the point you thought was not in the set.

6) Summarize your findings.

TURN IN ALL 3 STEPS PAPER-CLIPPED together in order.

Extra Credit: Create your own color map.

http://www.nahee.com/spanky/www/fractint/fractint.html - for info on Fractint


Unit 5: Graphs of Functions – Revisited

Solving Equations Graphically

The Intersection Method

To solve an equation of the form ( ) ( )f x g x :

1. Graph 1 ( )y f x and 2 ( )y g x on the same screen.

2. Find the x - coordinate of each point of intersection.

Ex. 1: Solve.

a) 2 1

43

x

x

b) 2 3 4x x

c) 2 34 3 6x x x x

The x - intercept Method

To solve an equation of the form ( ) ( )f x g x :

1. Write the equation in the equivalent form ( ) 0f x .

2. Graph ( )y f x .

3. The x - intercepts of the graph are the real solutions to the equation.

Ex.2: Solve.

a) 2 1

43

x

x

b) 2 3 4x x

c) 2 34 3 6x x x x

d) 5 2 3 5x x x


Technological Quirks

1. Solve ( ) 0f x by solving ( ) 0f x .

2. Solve ( )

0( )

f x

g x by solving ( ) 0f x (eliminate any values that also make ( ) 0g x ).

Ex. 3: Solve.

a) 4 2 2 1 0x x x b)

2

2

2 10

9 9 2

x x

x x

Applications

Ex. 1: According to data from the U.S. Bureau of the Census, the approximate population y (in millions)

of Chicago and Los Angeles between 1950 and 2000 are given by:

Chicago: 3 2.0000304 .0023 .02024 3.62y x x x

Los Angeles: 3 2.0000113 .000992 .0538 1.97y x x x

where 0x corresponds to 1950. In what year did the two cities have the same population?

Ex. 2: The average of two real numbers is 41.125, and their product is 1683. Find the two numbers.

Ex. 3: A rectangle is twice as wide as it is high. If it has an area of 24.5 square inches, what are the

dimensions of the rectangle?


Ex. 4: A rectangular box with a square base and no top is to have a volume of 30,000 3cm . If the surface

area of the box is 6000 2cm , what are the dimensions of the box?

Ex. 5: A box with no top that has a volume of 1000 cubic inches is to be constructed from a 22 x 30-inch

sheet of cardboard by cutting squares of equal size from each corner and folding up the sides. What size

square should be cut from each corner?

Ex. 6: A pilot wants to make 840-mile trip from Cleveland to Peoria and back in 5 hours flying time.

There will be a headwind of 30 mph going to Peoria, and it is estimated that there will be a 40-tail wind

on the return trip. At what constant engine speed should the plane be flown?


Solving Inequalities Graphically

1. Rewrite the inequality in the form ( ) 0f x or ( ) 0f x .

2. Determine the zeros of f .

3. Determine the interval(s) where the graph is above ( ( ) 0f x ) or below ( ( ) 0f x ) the x -axis.

Ex. 1: Solve each inequality graphically. Express your answer in interval notation.

a) 2( 4)( 3) 0x x x

b) 2 3 4x x

c) 2

30

4

x

x d)

3 2

4 1x x

e)

f) 4 3 26 2 5 2x x x x

Ex. 2: A company store has determined the cost of ordering and storing x laser printers is:

300,0002c x

x

If the delivery truck can bring at most 450 printers per order, how many printers should be ordered at a

time to keep the cost below $1600.00?


Increasing, Decreasing and Constant Functions

A function f is increasing on an interval when, for any 1x and 2x in the interval, 1x < 2x implies

1 2( ) ( ).f x f x

A function f is decreasing on an interval when, for any 1x and 2x in the interval, 1x < 2x implies

1 2( ) ( ).f x f x

A function f is constant on an interval when, for any 1x and 2x in the interval, 1 2( ) ( ).f x f x

Ex.1: Determine the open intervals on which each function is increasing, decreasing or constant.

a) ( ) 1 3f x x x b) 3( ) 3f x x x c) 3( )f x x


Relative Minimum and Maximum Values (Relative Extrema)

A function value ( )f a is called relative minimum of f when there exists an interval 1 2( , )x x

that contains a such that 1 2x x x implies ( ) ( ).f a f x

A function value ( )f a is called relative maximum of f when there exists an interval 1 2( , )x x

that contains a such that 1 2x x x implies ( ) ( ).f a f x

Ex. 2: Determine the relative minimum and x -intercepts of 2( ) 3 4 2f x x x

Ex. 3: Use a graphing utility to determine the relative minimum and x -intercepts of 2( ) 3 4 2f x x x

Ex. 4: Use a graphing utility to determine any relative minima or maxima for 3( )f x x x


Ex. 5: During a 24-hour period, the temperature ( )t x (in degrees Fahrenheit) of a certain city can be

approximated by the model 3 2( ) .026 1.03 10.2 34t x x x x , 0 24x

where x represents the time of day, with 0x corresponding to 6 A.M.

Approximate the maximum and minimum temperatures during this 24-hour period.

Optimization: Translating Words into Functions – revisited

Ex.1: The sum of two nonnegative numbers is 15. Express the product of one and the square of the

other as a function of one of the numbers. Use a graphing utility to find the maximum product.

Ex.2: A rectangle has an area of 400 2in . Express the perimeter of the rectangle as a function of the

length of one of its sides. Use a graphing utility to find the minimum perimeter.

Ex. 3: An open box is made from a rectangular piece of cardboard that measures 30cm by 40cm by

cutting a square of length x from each corner and bending up the sides. Express the volume of the box

as a function of x . Use a graphing utility to find the dimensions of the box with the maximum volume.


Ex. 4: Express the area of the rectangle as a function of x . The equation of the line is 2 4x y .The

lower left-hand corner is on the origin and upper right-hand corner of the rectangle with coordinate

( , )x y is on the line. Use a graphing utility to find the rectangle with the maximum area.

Concavity and Inflection Points

Concavity is used to describe the way a curve bends. For any two points in a given interval that lie on a

curve, if the line segment that connects them is above the curve, then the curve is said to be concave up

over the given interval. If the segment is below the curve, then the curve is said to be concave down

over the interval. A point where the curve changes concavity is called an inflection point.


Ex. 1 For the following functions, estimate the following:

1. All local maxima and minima (relative extrema) of the function

2. Intervals where the function is increasing and/or decreasing

3. All inflection points of the function

4. Intervals where the function is concave up and when it is concave down

a) 3 2( ) 2 6 3f x x x x b) 3( ) 4 2g x x x

c) 2

3( )

( 2)f x

x

d) 2

( )1

x xf x

x


Unit 6: Trigonometry – Part1

Right Triangle Trigonometry

Hypotenuse Opposite

Adjacent

a) Sine

sin( )

b) Cosine

cos( )

c) Tangent

tan( )

d) Cosecant

csc( )

e) Secant

sec( )

f) Cotangent

cot( )

Ex. 1: Find the values of the six trigonometric functions of the angle .

3

Ex.2: Find the exact values of the sin,cos, and tan of 45

7

45˚


Ex. 3: Find the exact values of the sin,cos, and tan of 60 and 30

Ex. 4: Find the exact value of x (without a calculator).

5

x

Ex. 5: Find all missing sides and angles (with a calculator).

21

30˚

33˚

30˚


Applications

An angle of elevation and an angle of depression can be measured from a point of reference and a

horizontal line. Draw two figures to illustrate.

Ex. 6: (use a calculator) A surveyor is standing 50 feet from the base of a large building. The surveyor

measures the angle of elevation to the top of the building to be 71.5˚. How tall is the building? Draw a

picture.

Ex. 7: A ladder leaning against a house forms a 67˚angle with the ground and needs to reach a window

17 feet above the ground. How long must the ladder be?


Angles – Degrees and Radians.

An angle consists of two rays that originate at a common point called the vertex. One of the rays is

called the initial side of the angle and the other ray is called the terminal side.

Angles that share the same initial side and terminal side are said to be coterminal.

To find a co-terminal angle to some angle (in degrees):

Radians – the other angle measure.

One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius

of the circle.


Conversion between degrees and radians

360 = radians.

Therefore,

a) 1 radian = degrees

b) 1 = radians

Ex. 1: Convert the following radian measure to degrees:

a) 5

6 b)

10

c) 4

d) 3

Ex. 2: Convert the following degree measure to radians:

a) 400 b) -120

To find a co-terminal angle to some angle (in radians):

Ex. 3: Find a coterminal angle, one positive and one negative, to 5

3.


We use the unit circle to quickly evaluate the trigonometric functions of the common angle found on it.

To summarize how to evaluate the Sine and Cosine of the angles found on the unit circle:

1. sin( ) =

2. cos( )=

Ex. 1: Find the exact value

a) sin( )4

b) cos( )2

c) sin( )6

d) cos( )3

e) sin(150 )

f) 11

cos( )6

g) 3

sin( )2

h) 5

cos( )3

i) sin(330 )

Ex. 2: Find the exact value by finding coterminal angles that are on the unit circle.

a) 13

sin( )4

b) 7

sin( )6

c) sin( 300 )


Cosine is an even function. Sine is an odd function.

cos( ) cos( ) sin( ) sin( )

Ex. 3: Find the exact value.

a) 7

cos( )6

b) 3

sin( )4

c) 13

cos( )4

Reference Angles

For any angle in standard position, the reference angle ( ' ) associated with is the acute angle

formed by the terminal side of and the x - axis.

Ex. 1: Find the reference angle ' for the given angles.

a) 2

3 b) 2.3 c)

5

4 d)

5

3


The signs (+ or – value) of the Sine, Cosine and Tangent functions in the four quadrants of the Euclidean

plane can be summarized in this way:

Ex.2: Find the exact value.

a) 2

sin( )3

b) 5

cos( )6

c) 5

sin( )3

d) cos( )3

e) cos( 300 ) f) sin(150 )

Ex.3: Find all values of in the interval [0,2 ] that satisfy the given equation

a) 2

sin( )2

b) 3

cos( )2

Ex. 5: If 2

sin( )3

t and 3

2t , find the value of cos( )t .


Arc Length

In a circle of radius r , the length s of an arc with

angle radians is: s r

Ex. 1: Find the length of an arc of a circle with radius 5 and an angle5

4.

Ex. 2: Find the length of an arc of a circle with radius 13 and an angle30 .

Ex. 3: The arc of a circle of radius 3 associated with angle has length 5. What is the measure of ?

Area of a Circular Sector

In a circle of radius r , the area A of a circular sector formed by an angle of radians is

21

2A r

Ex. 1: Find the area A of a sector with angle 45 in a circle of radius 4.


Graphs of the Sine and Cosine Functions

Ex. 1: Graph ( ) sinf x x

Domain: Range: x -intercepts: Period: Amplitude: Even or odd?

Ex. 2: Graph ( ) cosf x x

Domain: Range: x -intercepts: Period: Amplitude: Even or odd?


Ex.3: Graph one period of each function

a) ( ) 2cos( )f x x b) ( ) 1 sin( )f x x

c) ( ) cos( )2

f x x

d) ( ) sin(2 )f x x


Graphs of ( ) sin( )f x A Bx C Dand ( ) cos( )f x A Bx C D where 0A and 0B have:

Amplitude:

Period:

Horizontal shift (Phase shift):

Vertical shift:

Ex.1: Graph one period of each function.

a) ( ) 2sin( )3

f x x

Amplitude: Period: Horizontal Shift: End Points:

b) 1

( ) 3sin( )2 4

f x x



c) ( ) 1 2cos(2 )4

f x x


Ex. 2: Write a Sine or Cosine function whose graph matches the given curve.

a) x -scale is 4

b) x -scale is

Ex. 3: Write a Sine and Cosine function whose graph matches the given curve. x -scale is 3


Unit 7: Trigonometry – Part2 Revisiting Tangent, Cotangent, Secant and Cosecant These are called the Quotient Identities:

a) sin( )

tan( )cos( )

b) cos( )

cot( )sin( )

The following are called the Reciprocal Identities:

a) 1

csc( )sin( )

b) 1

sec( )cos( )

c) 1

cot( )tan( )

d) 1

tan( )cot( )

e) 1

sin( )csc( )

f) 1

cos( )sec( )

Ex. 1 Evaluate all six trigonometric functions at the following values of :

a) 6

b) 2


The Pythagorean Identities:

a) 2 2sin ( ) cos ( ) 1x x

Using the Quotient and Reciprocal identities we can derive the other two Pythagorean Identities:

2 2sin ( ) cos ( ) 1x x 2 2sin ( ) cos ( ) 1x x

Conclusion – The other two identities are:

b) c) Ex. 2 Find the values of all six trigonometric functions from the given information:

a) 4

sin( ) ,5

is in the first quadrant. b) csc( ) 5 , 3

22


Graphs of the Tangent and Cotangent Functions

Ex. 1 Graph ( ) tanf x x

Domain: Range: x -intercepts: Period: Even or odd?

Ex. 2 Graph ( ) cotf x x



Ex.3 Sketch one period of each function

e) ( ) 2tan( )f x x f) ( ) cot( )

4f x x

Ex. 4 Find the period of the following functions:

( ) tan(2 )f x x ( ) tan( )

2f ( ) cot( )

3

xf x

Ex. 5 Find all the values of t in the interval [0,2 ] satisfying the given equation:

a) tan( ) 1 0t b) cot( ) 3 0t


Graphs of the Secant and Cosecant Functions

Ex. 1 Graph ( ) secf x x


Ex. 2 Graph ( ) cscf x x



Ex.3 Graph one period of each function

g) 1

( ) sec( )2 4

f x x h) ( ) 2csc( )2

f x x

More on Trigonometric Identities Ex. 1 Use the identities you have learned so far to verify the following:

a) 2 3sin( )cos ( ) sin( ) sin ( ) b) 2 2 2(1 (cos ) )(sec ) (tan )x x x


c) 21 2sin cos (sin cos ) d) cot tan sec cscx x x x

Sum and Difference formulas for Sine and Cosine

sin( ) sin cos cos sin

sin( ) sin cos cos sin

cos( ) cos cos sin sin

cos( ) cos cos sin sin

Ex. 2 Use the sum and difference formulas to determine the value of the following trigonometric functions.

a) 3

sin( )6 4

b) 7

cos( )12

Ex. 3 Verify the identity:

sin( ) cos2

t t


We use the sum and difference formula to derive the double angle formulas.

1. sin(2 )x 2sin( )cos( )x x

2. 2 2cos(2 ) cos ( ) sin ( )x x x

Verification: We can then use the Pythagorean identities to derive two other versions of the double angle formula for cosine.

3. 2cos(2 ) 1 2sin ( )x x

4. 2cos(2 ) 2cos ( ) 1x x

Verification: Power-Reducing formulas

If we solve for 2sin ( )x and 2cos ( )x in #3 and #4 above, we get:

a) 2 cos(2 ) 1cos ( )

2

xx b) 2 1 cos(2 )

sin ( )2

xx

These formulas should be memorized and are very useful in integral calculus.


Ex. 4 If3

sin5

t , 3

2t ¸ find cos(2 ),sin(2 )t t and tan(2 )t

Ex. 5 Verify the identities

a) cos sin sin2 cos cos2x x x x x b) cot tan 2cot 2x x x Ex. 6 Find all the values of x in the interval [0,2 ] that satisfy the given equation.

a) sin2 sinx x b) 2(cos ) 3sin 3 0x x


Ex. 7 Find all values of t that satisfy the given equation.

a) 2cos 2t b) 1

sin2

t

c) 2(cos ) cos 0t t

d) 22sin sin 1 0t t

e) 3

sin(3 )2

t

f) csc 2t

g) 3tan(2 ) 3 0t

h) sin cost t


Unit 8: Inverse Trigonometric Functions

Inverse Trigonometric Functions The Inverse Sine Function In order for the sine function to have an inverse that is a function, we must first restrict its domain to

[ , ]2 2

so that it will be one-to-one and therefore have an inverse that is a function.

sin( )y x 1sin ( )y x

Domain: [ , ]2 2

Range:

Domain: Range:

The range of the arcsine function can be visualized by:

The arcsine function ( arcsin ( x )), or inverse sine function ( 1sin ( )x ), is defined by

arcsin( )y x iff sin( )x y where 1 1x and 2 2

y .

In other words, the arcsine of the number x is the angle y where 2 2

y whose sine is x .


Ex. 1 Find the exact value of the given expression.

a) arcsin (1

2) b) 1sin (

3

2)

c) 1sin (-1)

d) arcsin (2

2)

e) 3

arcsin(sin( ))4

f) 1

cos(arcsin( ))2

The Inverse Cosine Function In order for the cosine function to have an inverse that is a function, we must first restrict its domain to

[0, ] .

cos( )y x 1cos ( )y x

Domain: [0, ] Range:

Domain: Range:

The range of the arccosine function can be visualized by:


The arccosine function ( arccos ( x )), or inverse cosine function ( 1cos ( )x ), is defined by

arccos( )y x iff cos( )x y where 1 1x and 0 y .

In other words, the arccosine of the number x is the angle y where 0 y whose cosine is x .


a) arccos (1

2) b) 1cos (

3

2)

c) 1cos (-1)

d) arccos (2

2)

e) 5

arccos(cos( ))4

f) 1

cos(arcsin( ))3

The Inverse Tangent Function In order for the Tangent function to have an inverse that is a function, we must first restrict its domain

to ( , )2 2

.

tan( )y x 1tan ( )y x

Domain: ( , )2 2

Range:

Domain: Range:


The range of the arctangent function can be visualized by:

The arctangent function ( arctan ( x )), or inverse tangent function ( 1tan ( )x ), is defined by

arctan( )y x iff tan( )x y where 1 1x and 2 2

y .

In other words, the arctangent of the number x is the angle y where 2 2

y whose tangent is x .


a) 1tan (1)

b) arctan( 3)

c) arctan(tan )7

d) 1

sin(arctan( ))4

Ex. 4 Write the given expression as an algebraic expression in x .

a) 1sin(tan )x


“Algebraic” solutions to Trigonometric Equations Solutions for basic Trigonometric equations.

1. cos( )x c , ( 1 1c )

Solve : cos .6x

Solve : 8cos 1 0x

1cos ( ) 2x c n and 1cos ( ) 2x c n

2. sin( )x c , ( 1 1c )

Solve : sin .75x

Solve: 23sin sin 2 0x x

1sin ( ) 2x c n and 1( sin ( )) 2x c n


3. tan( )x c

Solve: tan 3.6x

Solve: 2sec 5tan 2x x

1tan ( )x c n

Angle of inclination

If L is a nonvertical line with angle of inclination (0 180 ), then tan = the slope of L .

Ex. 1 Find the angle of inclination of a line of slope5

3.

Ex. 2 Find the angle of inclination of a line of slope -2.


Law of Sines and Law of Cosines – techniques for solving general triangles. When we are given two angles and an included side (ASA), two angles and a non-included side (AAS), or two sides and a non-included angle (SSA), we can find the remaining sides and angles using the Law of Sines. Law of Sines

sin sin sin

a b c

Ex1. A telephone pole makes an angle of 82 with the ground. The angle of elevation of the sun is 76 . Find the length of the telephone pole if its shadow is 3.5m. (assume that the tilt of the pole is away from the sun and in the same plane as the pole and the sun). SSA – The ambiguous case. When given two sides and a non-included angle, there are three different scenarios:

a) No triangle b) One, unique triangle c) Two different triangles (since you will be solving for an angle with SSA, see if another triangle is

possible by subtracting the acute angle found with arcsine from 180 )


Ex. 2 Solve the triangle: 2, 1, 50a c

Ex. 3 Given a triangle with a = 22 inches, b =12 inches and =35 , find the remaining sides and angles.

Ex. 4 Solve the triangle: 6, 8, 35a b .


Law of Cosines We use the law of cosines when we are given three sides (SSS) or two sides and an included angle (SAS).

2 2 2 2 cosa b c bc 2 2 2 2 cosb a c ac 2 2 2 2 cosc a b ab

Ex. 5 Find all the missing angles of a triangle with sides 8, 19, 14a b c .

Ex. 6 A ship travels 60 miles due east and then adjusts its course 15 northward. After traveling 80 miles in that direction, how far is the ship from its departure?


Unit 9: Sequences and Series

A sequence is an ordered list of numbers and is formally defined as a function whose domain is the set

of positive integers.

It is common to use subscript notation rather than the standard function notation.

For example, we could use 1a to represent the first term of the sequence, 2a to represent the second

term, 3a the third term and so forth where na would represent the “nth term”

Note: Occasionally, it is convenient to begin a sequence with something other than 1a , such as 0a , in

which case the sequence would be 0a , 1a , 2a , 3a , … , 2na, 1na

, na, 1na , … (note how 1na

is the term

before na and so forth)

Ex. 1 List the first 5 terms of:

a) 3 ( 1)n

na

b) 1 2

n

nb

n

c) 2

2 1n n

nc

d) The recursively defined sequence 1 15, 25n na a a


The symbol !n (read “ n factorial”) is defined as ! ( 1) ( 2) 4 3 2 1n n n n where 0! 1.

For example, 6! 6 5 4 3 2 1 720

Ex. 2 Simplify the ratio of factorials

a) 25!

23!

b) ( 2)!

!

n

n

c) (2 )!

(2 2)!

n

n

Pattern Recognition for sequences

Ex.3 Find a sequence na whose first five terms are

a) 2 4 8 16 32

, , , , , ...1 3 5 7 9

b) 2 8 26 80 242

, , , , , ...1 2 6 24 120

c) 2 3 4

1, , , , , ...2 6 24

x x xx

8.1 1-85 e.o.o


Series

Definition of a Series

Consider the infinite sequence 1a , 2a , 3a , … ia , …

1. The sum of the first n terms of the sequence is called a finite series or the partial sum of the

sequence and is denoted by

1 2 3

1

...n

n i

i

a a a a a

where i is called the index of summation, n is the upper limit of summation and 1 is the lower

limit of summation

2. The sum of all the terms of an infinite sequence is called an infinite series and is denoted by

1 2 3

1

... ...i i

i

a a a a a

Ex.3 Find the sum:

a)

5

1

3i

i

b)

62

3

(1 )k

k

c)

8

0

1( )

!n n


d) 3

1

3( )10i

i

Ex. 4 Use sigma notation to write the sum 1 1 1 1

3(1) 3(2) 3(3) 3(9)

8.1 87-117

Arithmetic Sequences and Partial Sums

What behavior do the first two sequences have that the third one does not?

1. 7, 11, 15, 19, ….

2. 2, -3, -8, -13, -18 …

3. 1, 4, 9, 16, …

A sequence is arithmetic when the difference between consecutive terms is constant. We call this

difference the common difference ( d ) where 1n nd a a .

Ex.1 Find a formula for the nth term of the arithmetic sequence 7, 11, 15, 19, …. where 1a is the first

term.

The nth term of an arithmetic sequence has the form

1 ( 1)na a n d where d is the common difference and 1a is the first term.

Ex.2 Find a formula for the nth term of the arithmetic sequence whose common difference is 3 and

whose first term is 2. List the first five terms of this sequence.


Ex.3 The fourth term of an arithmetic sequence is 20, and the 13th term is 65. Write the first several

terms of this sequence.

Ex. 4 Find the ninth term of an arithmetic sequence whose first two terms are 2 and 9.

8.2 1-69 e.o.o.

The sum of a finite arithmetic sequence with n terms is given by:

1( )2

n n

nS a a

Note:

Ex.5 Find the sum: 1+3+5+7+9+11+13+15+17+19

Ex.6 Find the 150th partial sum of the sequence 5,16,27,38,49, ….

Ex. 7 500

1

(2 8)n

n


Ex. 8 50

0

(50 3 )n

n

8.2 71-81, 84

Geometric Sequences and Series

List the first five terms of the geometric sequence 2n

na

A sequence is geometric when the ratios of consecutive terms are constant. We call this constant the

common ratio(r) where 1

n

n

ar

a.

Ex.1 Determine if the following sequences are geometric and if so, determine the common ratio.

a) 12, 36, 108, 324, …

b) 1 1 1 1 1

, , , , , ...3 9 27 81 243

c) 1, 4, 9, 16, …..


The nth term of a geometric sequence has the form

1

1

n

na a r where r is the common ratio and 1a is the first term.

Ex. 2 Write the first five terms and the general term of the geometric sequence whose first term is

1 3a and whose common ratio is 2r .

Ex. 3 Find the 15th term of the geometric sequence whose first term is 20 and whose common ratio is

1.05.

Ex. 4 Find a formula for the nth term of the following geometric sequence. What is the ninth term of the

sequence?

5, 15, 45…

Ex. 5 The fourth term of a geometric sequence is 125 and the 10th term is125

64. Find the 14th term

(assume all terms are positive).

8.3 1-45 e.o.o.


The nth partial sum of a geometric sequence with common ratio of r and first term of 1a is given by

1

1 1

1

1( )1

nni

n

i

rS a r a

r

Note:

Ex. 6 Find the sum:

a) 12

1

(4(.3) )n

n

b) 7

1

0

2n

n

The Sum of an Infinite Geometric Series (or simply Geometric Series)

If | | 1r , then the infinite geometric series has the sum

11

0

( )1

i

i

aS a r

r

If | | 1r , then the series does not have a sum and diverges to infinity.


Ex. 7 Find the sum:

a) 0

(4(.6) )n

n

b) 3 + 0.3 + 0.03 + 0.003 + …

8.3 55-93 e.o.o


Unit 10: Exponential and Logarithmic Functions Rational exponents

If b is a real number and n and m are positive and have no common factors, then n

mb = ( )m n nmb b

Laws of exponents

a)

b)

c)

d)

e)

f)

g)

Ex. 1 Simplify

a) 1

38

( )27

b) 5

29

Exponential Function

If 0b and 1b , then an exponential function ( )y f x is a function of the form ( ) xf x b .

The number b is called the base and x is called the exponent.


Ex. 2 Graph each of the given functions

a) ( ) 2xf x and ( ) 4xf x

b) 1

( ) 2 ( )2

x xf x and 1

( ) 4 ( )4

x xf x


In general …

( ) xf x b

Domain: Range: Intercept: Horizontal Asymptote:

1( ) ( )x xf x b

b

Domain: Range: Intercept: Horizontal Asymptote:

Ex. 3 Sketch each of the given functions

a) 1( ) 3xh x

b) ( ) 5 2xh x


The Natural Base e

Use you calculator to explore 1

lim(1 )n

n n.

Conclusion: Ex. 4 Sketch each of the given functions

a) ( ) xf x e

b) ( ) xh x e

c) ( ) 2 xf x e

For c) State the Domain and Range


Ex. 5 Solve

a) 3 12 8x x

b) 2( 1)7 343x

c) 2 3 24 2x x d) 64 10(8 ) 16 0x x

Compound Interest

The amount of money ( )A t at some time t (in years) in an investment with an initial value, or principle

of P with an annual interest rate of r (APR – given as a decimal), compounded n times a year is:

( ) 1

ntr

A t Pn

Ex. 6 Determine the value of a CD in the amount of $1000.00 that matures in 6 years and pays 5% per year compounded

a) Annually

b) Monthly

c) Daily


Continuously Compounded Interest.

If the interest is compounded continuously ( n ), then the amount of money after t years is:

( ) rtA t Pe

Ex. 7 Determine the amount in the CD from example 6 if the interest is compounded continuously. Ex. 8 Which interest rate and compounding period gives the best return?

a) 8% compounded annually

b) 7.5% compounded semiannually

c) 7% compounded continuously Ex. 9 What initial investment at 8.5 % compounded continuously for 7 years will accumulate to $50,000?


Logarithmic Functions Set up

Sketch ( ) 2xf x . Give the domain and range. Then find 1( )f x .

1( )f x =

Domain: Range: Intercept: V.A.: Definition

For each positive number 0a and each x in (0, ) , logay x if and only if yx a . yx a is the corresponding “exponential form” of the given “logarithmic form” logay x .

Ex. 1 Evaluate each expression.

a) 10log 1000

b) 10log 0.1

c) 2log 32

d) 2log 4

e) 8log 8

f) 3log 1

g) 2

5log 5 h) 3log 83


Properties of the Logarithm function with base a .

a) log 1 0a b) log 1a a

c) log x

a a x

d) loga xa x

Ex. 2 On the same coordinate plane, sketch the following functions.

( ) 3xf x and 3( ) logg x x

1( ) ( )

2

xf x and 1/2( ) logg x x

In general ….

( ) logag x x , 1a

Domain: Range: Intercept: V.A.:

( ) logag x x , 0 1a

Domain: Range: Intercept: V.A.:


Ex. 3 Sketch the following functions.

3( ) log ( 2)g x x

1/2( ) log 1g x x

The Natural Logarithm Function

The function defined by ( ) log lnef x x x and lny x iff yx e .

Ex. 4 On the same coordinate plane, sketch the following functions.

( ) xf x e and ( ) lng x x


Properties of the Logarithm function with base e .

a) b)

c)

d)

Arithmetic Properties of Logarithms

For each positive number 1a , each pair of positive real numbers U and V , and each real number n

we have:

Base a Logarithm Natural Logarithm

a)

a) b)

b)

c)

c)

Ex. 5 Evaluate each expression

a) 4lne b) ln45e

c) 1

lne

d) (1/2)ln16e

e) 3ln8e

f) 2 2 2log 6 log 15 log 20

Change-of-Base Formula

For 0, 0, 0a a x ... log ln

loglog ln

a

x xx

a a

Ex. 6 Use your calculator to evaluate 6log 13 .


Ex.7 Use the properties of logarithms to simplify each expression so that the ln y does not contain

products, quotients or powers.

a) (2 1)(3 2)

4 3

x xy

x

b) 6 3 264 1 2y x x x

Solving Exponential and Logarithmic Equations

Ex. 8 Solve each of the given equations

a) 83xe b) 24 7xe

c) 2 13x

d) 2ln(3 ) 6x

e) ln( 1) ln( 3) 1x x

f) ln( 2) ln(2 3) 2lnx x x


g) 2log ( 3) 4x h) 1 5 32x xe

i) 2 6x x xe x e x e

j) ln 0x x x

Ex. 9 Given the function 3 1( ) 5xf x e , Find 1( )f x and state the domain and range of the inverse

function.


Exponential Growth and Decay

In one model of a growing (or decaying) population, it is assumed that the rate of growth (or decay) of

the population is proportional to the number present at time t (rate of growth = ( )kP t ). Using calculus,

it can be shown that this assumption gives rise to:

0( ) ktP t Pe where k is the rate of growth ( 0k ) or decay ( 0k ).

Ex.1 The number of a certain species of fish is given by 0.012( ) 12 tn t e where t is measured in years and

( )n t is measured in millions.

a) What is the relative growth rate of the population?

b) What will the fish population be after 15 years?

Ex.2 A bacteria culture starts with 500 bacteria and 5 hours later has 4000 bacteria. The population

grows exponentially.

a) Find a function for the number of bacteria after t hours.

b) Find the number of bacteria that will be present after 6 hours.

c) When will the population reach 15000?


Ex. 3 A culture of cells is observed to triple in size in 2 days. How large will the culture be in 5 days if the

population grows exponentially?

Ex. 4 Carbon-14, one of the three isotopes of carbon, has a half-life of 5730 years. If 10 grams were

present originally, how much will be left after 2000 years? When will there be 2 grams left?

Ex. 5 On September 19th, 1991, the remains of a prehistoric man were found encased in ice near the

border of Italy and Switzerland. 52.4% of the original carbon 14 remained at the time of the discovery.

Estimate the age of the Ice Man.


Ex. 6 The radioactive isotope strontium 90 has a half-life of 29.1 years.

a) How much strontium 90 will remain after 20 years from an initial amount of 300 kilograms?

b) How long will it take for 80% of the original amount to decay?

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