AP Precalculus - Unit 1: Polynomial and Rational Functions Flashcards

__Function:__ a mathematical relationship that maps a set of input values to a set of output values such that each input value is mapped to exactly one output value

Input Values = Domain = Independent Variable (x)

Output Values = Range = Dependent Variable (y)

The input and output of a function vary according to the __function rule__

This can be represented graphically, verbally, analytically, or numerically

__A function is increasing over an interval of its domain if__:

as the input values increase, the output values __always__ increase / for all a and b in the interval, if a < b, then f(a) < f(b)

__A function is decreasing over an interval of its domain if:__

as the input values increase, the output values __always__ decrease / for all a and b in the interval, if a < b, then f(a) > f(b)

Graph → a visual display of input-output pairs & shows how values vary

Concave up → a rate of change is increasing

Concave down → a rate of change decreasing

x-intercepts = zeros of the function

Average Rate of Change: The average rate of change over the closed interval [a, b] is the __slope of the secant line__ from the point (a, f(a)) to (b, f(b))

**The Rate of Change of a Function at a Point** → approximated by the average rate of change over small intervals containing the point

__Positive__ Rate of Change → when one quantity increases, the other quantity increases as well

__Negative__ Rate of Change → when one quantity increases, the other quantity decreases

AROC of Linear Function → constant

AROC changing at a rate of 0

AROC of Quadratic Function → slope of secant line (linear)

AROC changing at a constant rate

If AROC is increasing over interval → concave up

If AROC is decreasing over interval → concave down

Polynomial Functions:

n = postitive integer polynomial degree = n

a_{i} = a real number for each I from 1 to n leading term = a_{n}x^{n}

a_{n} → nonzero leading coefficient = a_{n}

**Local/Relative Maximum/Minimum**__:__ when polynomial changes between increasing and decreasing/included endpoint with __restricted domain__

**Global/Absolute Maximum/Minimum:** the greatest local maximum/least local minimum

Between two zeros of a polynomial → at least one extrema

**Points of Inflection:** when rate of change of function changes from increasing to decreasing or from decreasing to increasing; changes concavity

Polynomial with even degree → global maximum or global minimum

**Complex Numbers:** real numbers and non-real numbers

If p(a) = 0, then:

· a → zero/root of function p

· x-intercept at (a,0)

· if a is a real number, then (x – a) is a linear factor of function p

If linear factor (x – a) is repeated n times, then there are n zeros in the functions

└ a polynomial function of degree n has exactly n complex zeros

Real zeros of a polynomial can be endpoints for inequality intervals

If the real zero, a, has an even multiplicity (ex. (x – a)^{2}), then the graph with “bounce” off the x-axis at x = a

**Non-Real Zeros:** if a + bi is a non-real zero of a polynomial p, then its conjugate a – bi is also a zero of polynomial p

The degree of a polynomial function can be found by examining inputs and outputs of the function __only if the input values are over equal intervals__

└ The degree of the function = the least value of n for which the successive nth differences are constant

Graph of even function à symmetric over the line x = 0, f(-x) = x

**Even function**: when n is even in polynomial of the form p(x) = a_{n}x^{n}*

Graph of odd function à symmetrix over the point (0, 0), f(-x) = - f(x)

**Odd Function**: when n is odd in polynomial of the form p(x) = a_{n}x^{n}*

* where n ≥ 1 and a_{n} ≠ 0

__End Behavior__

When input values of a function __increase without bound__, output values will either:

Increase without bound

Decrease without bound

When input values of a function __decrease without bound__, output values will either:

Increase without bound

Decrease without bound

The degree and sign of the leading term of a polynomial determines the end behavior of the polynomial function

└ as input values increase/decrease without bound, the values of the leading term dominate

**Rational Function:** the ratio of two polynomials where the polynomial in the denominator ≠ 0

end behavior → affected by the polynomial of greater degree (values will dominate); can be understood by examining quotient of polynomial leading terms

ex.

x grows as at a faster rate than 3, so function will go to ∞

-if **numerator** dominates → quotient of leading terms is __nonconstant polynomial__ à og function shares same end behavior

└ if leading term polynomial is linear → graph of rational function has __slant asymptote parallel to the the graph of the line__

-if **denominator** dominates → quotient of leading terms is __(constant)/(nonconstant polynomial__) à graph of og function has __horizontal asymptote as y = 0__

-if **neither** dominates → quotient of leading terms is__ constant = horizontal asymptote of og function__

**Rational Function Real Zeros** → real zeros of the numerator in domain

└ real zeros of both polynomials in rational function are endpoints/asymptotes for intervals satisfying the inequalities r(x)≥ 0 or r(x) ≤ 0

**Vertical Asymptote** → zeros of the polynomial in the denominator (and not numerator) / the zero appears more times in the denominator than in the numerator

Ex.

since (x – 1) appears more times in the denominator, it would be a vertical asymptote

If input values__ are greater than asymptote__, then

or

If input values are __less than asymptote__, then

or

** **

**Hole** → when a zero appears more times in the numerator than the denominator

-Find the location of the hole by plugging in zero value into function

__standard form__ of polynomial and rational functions à used to find end behavior

__factored form__ of polynomial and rational functions à used to find x-intercepts, asymptotes, holes, domain, and range

**Polynomial Long Division** à used to find equations of slant asymptotes of graphs of rational functions

└ degree of remainder is less that degree of divider

**Binomial Theorem** → used to expand terms in the form (a + b)^{n} and polynomials functions in the form of (x + c)^{n} (where c is a constant) by using Pascal’s Triangle

Ex.

**Pascal’s Triangle:**

Functions can be transformed from parent function f(x)

g(x) = f(x) + k → **vertical transformation** of f(x) by k units

g(x) = f(x + h) → **horizontal transformation** of f(x) by -h units

g(x) = a f(x), where a ≠ 0 → **vertical dilation** by a factor of |a|;

if a < 0 → **reflection** over the x-axis

g(x) = f(bx), where b ≠ 0 → **horizontal dilation** by a factor of | |, if b < 0 → **reflection** over y-axis

*The domain and range of a transformed function may differ from the parent function

**Linear Functions** → used for contextual scenarios with roughly __constant rates of change__

**Quadratic Functions** → used for contextual scenarios with roughly __linear rates of change/__ roughly __symmetrical data sets__ with a unique minimum/maximum value

**Geometric** → used for contextual scenarios involving __area/volume__; two dimensions modeled by quadratic functions, three dimensions modeled by cubic functions.

**Polynomial** → used to model data sets/scenarios with __multiple zeros or multiple extremas__

**Piece-wise** → a set of functions defined over nonoverlapping domain interval; used for data sets/contextual scenarios that have __different characteristics over different intervals__

__Assumptions and Restrictions of Function Model__

a model may:

-assume what is consistent

-how quantities change together

-require domain restrictions (based on mathematical clues, context clues and/or extreme values)

-require range restrictions (ex. rounding values) (based on mathematical clues, context clues and/or extreme values)

A model can be constructed in multiple ways:

- based on **restrictions** from a scenario

- using **transformation**s from the parent function

- technology and **regressions** (linear, quadratic, cubic, & quartic)

- a **piece-wise function** can be constructed through a combination of the techniques above

Data sets that have quantities that are inversely proportional → __rational functions__

Ex.

__Application:__

Models can be used to draw conclusions about the data set/scenario

Appropriate units of measure should be used when given

__Function:__ a mathematical relationship that maps a set of input values to a set of output values such that each input value is mapped to exactly one output value

Input Values = Domain = Independent Variable (x)

Output Values = Range = Dependent Variable (y)

The input and output of a function vary according to the __function rule__

This can be represented graphically, verbally, analytically, or numerically

__A function is increasing over an interval of its domain if__:

as the input values increase, the output values __always__ increase / for all a and b in the interval, if a < b, then f(a) < f(b)

__A function is decreasing over an interval of its domain if:__

as the input values increase, the output values __always__ decrease / for all a and b in the interval, if a < b, then f(a) > f(b)

Graph → a visual display of input-output pairs & shows how values vary

Concave up → a rate of change is increasing

Concave down → a rate of change decreasing

x-intercepts = zeros of the function

Average Rate of Change: The average rate of change over the closed interval [a, b] is the __slope of the secant line__ from the point (a, f(a)) to (b, f(b))

**The Rate of Change of a Function at a Point** → approximated by the average rate of change over small intervals containing the point

__Positive__ Rate of Change → when one quantity increases, the other quantity increases as well

__Negative__ Rate of Change → when one quantity increases, the other quantity decreases

AROC of Linear Function → constant

AROC changing at a rate of 0

AROC of Quadratic Function → slope of secant line (linear)

AROC changing at a constant rate

If AROC is increasing over interval → concave up

If AROC is decreasing over interval → concave down

Polynomial Functions:

n = postitive integer polynomial degree = n

a_{i} = a real number for each I from 1 to n leading term = a_{n}x^{n}

a_{n} → nonzero leading coefficient = a_{n}

**Local/Relative Maximum/Minimum**__:__ when polynomial changes between increasing and decreasing/included endpoint with __restricted domain__

**Global/Absolute Maximum/Minimum:** the greatest local maximum/least local minimum

Between two zeros of a polynomial → at least one extrema

**Points of Inflection:** when rate of change of function changes from increasing to decreasing or from decreasing to increasing; changes concavity

Polynomial with even degree → global maximum or global minimum

**Complex Numbers:** real numbers and non-real numbers

If p(a) = 0, then:

· a → zero/root of function p

· x-intercept at (a,0)

· if a is a real number, then (x – a) is a linear factor of function p

If linear factor (x – a) is repeated n times, then there are n zeros in the functions

└ a polynomial function of degree n has exactly n complex zeros

Real zeros of a polynomial can be endpoints for inequality intervals

If the real zero, a, has an even multiplicity (ex. (x – a)^{2}), then the graph with “bounce” off the x-axis at x = a

**Non-Real Zeros:** if a + bi is a non-real zero of a polynomial p, then its conjugate a – bi is also a zero of polynomial p

The degree of a polynomial function can be found by examining inputs and outputs of the function __only if the input values are over equal intervals__

└ The degree of the function = the least value of n for which the successive nth differences are constant

Graph of even function à symmetric over the line x = 0, f(-x) = x

**Even function**: when n is even in polynomial of the form p(x) = a_{n}x^{n}*

Graph of odd function à symmetrix over the point (0, 0), f(-x) = - f(x)

**Odd Function**: when n is odd in polynomial of the form p(x) = a_{n}x^{n}*

* where n ≥ 1 and a_{n} ≠ 0

__End Behavior__

When input values of a function __increase without bound__, output values will either:

Increase without bound

Decrease without bound

When input values of a function __decrease without bound__, output values will either:

Increase without bound

Decrease without bound

The degree and sign of the leading term of a polynomial determines the end behavior of the polynomial function

└ as input values increase/decrease without bound, the values of the leading term dominate

**Rational Function:** the ratio of two polynomials where the polynomial in the denominator ≠ 0

end behavior → affected by the polynomial of greater degree (values will dominate); can be understood by examining quotient of polynomial leading terms

ex.

x grows as at a faster rate than 3, so function will go to ∞

-if **numerator** dominates → quotient of leading terms is __nonconstant polynomial__ à og function shares same end behavior

└ if leading term polynomial is linear → graph of rational function has __slant asymptote parallel to the the graph of the line__

-if **denominator** dominates → quotient of leading terms is __(constant)/(nonconstant polynomial__) à graph of og function has __horizontal asymptote as y = 0__

-if **neither** dominates → quotient of leading terms is__ constant = horizontal asymptote of og function__

**Rational Function Real Zeros** → real zeros of the numerator in domain

└ real zeros of both polynomials in rational function are endpoints/asymptotes for intervals satisfying the inequalities r(x)≥ 0 or r(x) ≤ 0

**Vertical Asymptote** → zeros of the polynomial in the denominator (and not numerator) / the zero appears more times in the denominator than in the numerator

Ex.

since (x – 1) appears more times in the denominator, it would be a vertical asymptote

If input values__ are greater than asymptote__, then

or

If input values are __less than asymptote__, then

or

** **

**Hole** → when a zero appears more times in the numerator than the denominator

-Find the location of the hole by plugging in zero value into function

__standard form__ of polynomial and rational functions à used to find end behavior

__factored form__ of polynomial and rational functions à used to find x-intercepts, asymptotes, holes, domain, and range

**Polynomial Long Division** à used to find equations of slant asymptotes of graphs of rational functions

└ degree of remainder is less that degree of divider

**Binomial Theorem** → used to expand terms in the form (a + b)^{n} and polynomials functions in the form of (x + c)^{n} (where c is a constant) by using Pascal’s Triangle

Ex.

**Pascal’s Triangle:**

Functions can be transformed from parent function f(x)

g(x) = f(x) + k → **vertical transformation** of f(x) by k units

g(x) = f(x + h) → **horizontal transformation** of f(x) by -h units

g(x) = a f(x), where a ≠ 0 → **vertical dilation** by a factor of |a|;

if a < 0 → **reflection** over the x-axis

g(x) = f(bx), where b ≠ 0 → **horizontal dilation** by a factor of | |, if b < 0 → **reflection** over y-axis

*The domain and range of a transformed function may differ from the parent function

**Linear Functions** → used for contextual scenarios with roughly __constant rates of change__

**Quadratic Functions** → used for contextual scenarios with roughly __linear rates of change/__ roughly __symmetrical data sets__ with a unique minimum/maximum value

**Geometric** → used for contextual scenarios involving __area/volume__; two dimensions modeled by quadratic functions, three dimensions modeled by cubic functions.

**Polynomial** → used to model data sets/scenarios with __multiple zeros or multiple extremas__

**Piece-wise** → a set of functions defined over nonoverlapping domain interval; used for data sets/contextual scenarios that have __different characteristics over different intervals__

__Assumptions and Restrictions of Function Model__

a model may:

-assume what is consistent

-how quantities change together

-require domain restrictions (based on mathematical clues, context clues and/or extreme values)

-require range restrictions (ex. rounding values) (based on mathematical clues, context clues and/or extreme values)

A model can be constructed in multiple ways:

- based on **restrictions** from a scenario

- using **transformation**s from the parent function

- technology and **regressions** (linear, quadratic, cubic, & quartic)

- a **piece-wise function** can be constructed through a combination of the techniques above

Data sets that have quantities that are inversely proportional → __rational functions__

Ex.

__Application:__

Models can be used to draw conclusions about the data set/scenario

Appropriate units of measure should be used when given