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 = sequences
they 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 extrema
g(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 = sequences
they 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 extrema
g(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