AP Pre Calculus Unit 2 *CRASH COURSE*

# 2.1 Arithmetic and Geometric Sequences

sequence: function from the whole numbers (n) to real numbers

graph consists of discrete points instead of curves because we are dealing with whole numbers

### Arithmetic

sequence of numbers with a common difference

(n)th term is given by a^1 + (n-1)d

d = the common difference

*CONSTANT RATE OF CHANGE*input = term number

output = term value

can be positive (increasing) or negative (decreasing)

### Geometric

sequence of numbers with a common ratio

(n)th term is given by g^1 ( r )^(n-1)

r = common ratio

*PROPORTIONAL CHANGE*can be positive (increasing) or negative (decreasing)

decreasing = r < 1 / Increasing = r > 1

# 2.2 Change in Linear & Exponential Functions

### Linear Functions (arithmetic)

f(x)= mx + b

the first term is represented by a^0

the roc is represented by d or “m”

must have a constant roc

point-slope formula and explicit formula are

*NOT*the same

### Exponential Functions (geometric)

f(x)= a(b)^x

the ratio of consecutive terms is the same

represented by g^n = g^0 ( r )^n

functions

*DO NOT*= sequencesthey may have different domains and ranges

# 2.3 Exponential Functions

x is always in the exponent

0 < x < 1 = exponential decay (concave down)

x > 1 = exponential growth (concave up)

domain always = all real numbers

output values are proportional over equal-length consecutive input values

*ALWAYS*decreasing or increasing*NO*points of inflection or extremag(x) = f(x) + k is an additive transformation (up by k units)

make sure to know the limit statements

# 2.4 Exponential Function Manipulation

## Properties

power to power means multiplying powers

multiplying means adding powers

negative exponent property means x^-n = 1/x^n *ONLY VALID FOR POSITIVE BASES*

exponential unit fraction- b^1/k ( k= natural numbers)

rational exponents- b^1/n = n√b

# 2.5 Exponential Function Context & Data Modeling

## Input - Output Pairs

can construct exponential functions

initial value = a / base= b

set up a system of equations ( or through an exponential regression) to find the best-fit function

correlation coefficient = r²: measures how well the data fits

residuals can help determine if the graph is appropriate

f(x) = e^x (continuous growth)

f(x) = e^-x (continuous decay)

# 2.6 Competing Function Model Validation

identify patterns

*remember to consider the domain, range, and purpose of the function*

## Appropriation

analyze residuals of regressions

no pattern means it is appropriate

predicted trends won’t always fit points (errors will happen)

Errors are the predicted value minus the actual value

# 2.7 Composite Functions

made up of 2 or more simpler functions put together

combining functions = composition of functions

basic notation: f(g(x))

*aka “f of g of x*”- output of the inside function is the input of the second function

1. identify inside and outside function

substitute each x with the inner function

# 2.8 Inverse Functions

### Criteria

the function must be a 1 to 1 ratio

the domain must not be restricting

notation = f^-1

domain and range will swap in an inverse

*NOT ALL FUNCTIONS HAVE AN INVERSE**INVERSES ARE NOT ALWAYS FUNCTIONS*

### Steps to Finding the Inverse

change f(x) to y

swap x and y roles

find inverse by solving for y

# AP Pre Calculus Unit 2 *CRASH COURSE*

# 2.1 Arithmetic and Geometric Sequences

sequence: function from the whole numbers (n) to real numbers

graph consists of discrete points instead of curves because we are dealing with whole numbers

### Arithmetic

sequence of numbers with a common difference

(n)th term is given by a^1 + (n-1)d

d = the common difference

*CONSTANT RATE OF CHANGE*input = term number

output = term value

can be positive (increasing) or negative (decreasing)

### Geometric

sequence of numbers with a common ratio

(n)th term is given by g^1 ( r )^(n-1)

r = common ratio

*PROPORTIONAL CHANGE*can be positive (increasing) or negative (decreasing)

decreasing = r < 1 / Increasing = r > 1

# 2.2 Change in Linear & Exponential Functions

### Linear Functions (arithmetic)

f(x)= mx + b

the first term is represented by a^0

the roc is represented by d or “m”

must have a constant roc

point-slope formula and explicit formula are

*NOT*the same

### Exponential Functions (geometric)

f(x)= a(b)^x

the ratio of consecutive terms is the same

represented by g^n = g^0 ( r )^n

functions

*DO NOT*= sequencesthey may have different domains and ranges

# 2.3 Exponential Functions

x is always in the exponent

0 < x < 1 = exponential decay (concave down)

x > 1 = exponential growth (concave up)

domain always = all real numbers

output values are proportional over equal-length consecutive input values

*ALWAYS*decreasing or increasing*NO*points of inflection or extremag(x) = f(x) + k is an additive transformation (up by k units)

make sure to know the limit statements

# 2.4 Exponential Function Manipulation

## Properties

power to power means multiplying powers

multiplying means adding powers

negative exponent property means x^-n = 1/x^n *ONLY VALID FOR POSITIVE BASES*

exponential unit fraction- b^1/k ( k= natural numbers)

rational exponents- b^1/n = n√b

# 2.5 Exponential Function Context & Data Modeling

## Input - Output Pairs

can construct exponential functions

initial value = a / base= b

set up a system of equations ( or through an exponential regression) to find the best-fit function

correlation coefficient = r²: measures how well the data fits

residuals can help determine if the graph is appropriate

f(x) = e^x (continuous growth)

f(x) = e^-x (continuous decay)

# 2.6 Competing Function Model Validation

identify patterns

*remember to consider the domain, range, and purpose of the function*

## Appropriation

analyze residuals of regressions

no pattern means it is appropriate

predicted trends won’t always fit points (errors will happen)

Errors are the predicted value minus the actual value

# 2.7 Composite Functions

made up of 2 or more simpler functions put together

combining functions = composition of functions

basic notation: f(g(x))

*aka “f of g of x*”- output of the inside function is the input of the second function

1. identify inside and outside function

substitute each x with the inner function

# 2.8 Inverse Functions

### Criteria

the function must be a 1 to 1 ratio

the domain must not be restricting

notation = f^-1

domain and range will swap in an inverse

*NOT ALL FUNCTIONS HAVE AN INVERSE**INVERSES ARE NOT ALWAYS FUNCTIONS*

### Steps to Finding the Inverse

change f(x) to y

swap x and y roles

find inverse by solving for y