Exam 3 Flashcards
zero of a polynomial
number “k” such that f(k) = 0
if k is a real number, it is an x-intercept
MUST be a factor of a polynomial
(x-k) where k is a zero of the polynomial
synthetic division can be used when…
dividing by (x - k)
used to get a factor from a polynomial that can’t be factored
rational zeros theorem
define rational zeros theorem
for any polynomial with a leading coeffecient of Q and a constant of P, any rational zero must =
factor of P / factor of Q
steps to rational zero theorem
- find Q (leading coeffecient) and P (constant)
- find factors of Q and P
- set up all possible divisions of a factor of Q by a factor of P, and eliminate duplicates
- plug into function until you find one that makes f(x) = 0
- make a factor from this zero and divide from polynomial
steps to rational zero theorem
- find Q (leading coeffecient) and P (constant)
- find factors of Q and P
- set up all possible divisions of a factor of P by a factor of Q (p/q), and eliminate duplicates
- plug into function until you find one that makes f(x) = 0
- make a factor from this zero and divide from polynomial
domain of a rational
set of all x-values such that the denominator =/= 0
how to find LCD for polynomials
out of a list of all factors of the polynomials in the denominators, multiply each unique factor, and raise factors to the highest power found on them in the denominators
what to check for after solving rational equations
must make sure x-value does not cause division by 0
how to find x-intercepts in rationals
set equal to 0
disregard denominator
find zeros of numerator
m. 1 effect on x-int
straight through
m. even effect on x-int
bounces off
m. odd (except 1) effect on x-int
S-shape
m. odd effect on vertical asymptote
opposite directions
m. even effect on vertical asymptotes
same direction (volcano or trench)
how to find VA
find x-values that make denominator = 0
horizontal asymptotes are part of ______ behavior
end
vertical asymptotes are part of ______ behavior
middle
equation for HA
y = #
how to find HA
leading coeffecient of numerator / leading coefficient of denominator
if # / x, then y = 0
if x / #, it is a slant asymptote
used to solve polynomial inequalities
analytical technique
analytic technique steps
- put 0 on one side of equation
- ID x-ints and VAs
- mark on a number line
- test values in each interval on the number line in the function
- solution is the union of the true intervals (VAs are never included, x-ints are if inequality includes =)
Ab/c =
c√Ab
denominator goes out; numerator stays in
necessary after solving radical equations
check work - there may be no solution if the square root of a number gives you a negative value!
define 1-to-1 function
function in which each y value connects to only one x value
1-1 functions must pass…
both vertical & horizontal line tests
if x leads to multiple y values…
not a function
if y leads to multiple x values…
not a 1-1 function
difference quotient =
f(x + h) - f(x) / h
if all is correct in DQ, h will…
cancel
2 exceptions to domain of all real numbers for functions
- asymptotes of rationals caused by division by 0
- square root functions, in which whatever is beneath the √ must be > or = 0
“slope” of nonlinear functions =
average rate of change =
f(b) - f(a) / b - a
define function extrema
minimum and maximum points on a graph
y-values only
infinity does not count
define absolute minimum and maximum
highest and lowest y-value on a graph that is not infinity
define local maximum and minimum
output of a function where the function switches from decreasing to increasing OR increasing to decreasing
switching to a constant interval does not count
find domain of a composed function
- find domain of the final composed function
- unite it with the domain of the inner function
find domain of a combined function
intersection of domains of original functions
define transformation
change to the function which keeps the core structure of the graph
all can be represented as an addition of, subtraction of, multiplication by, or division by a number
3 vertical (y-coordinate) transformations
vertical stretch/squish
reflection across the x-axis
shift up or down
3 horizontal (x-coordinate) transformations
horizontal stretch/squish
reflection across y-axis
shift left or right
VERTICAL STRETCH
change to equation
change to coordinate
y = f(x) * h
y * h
REFLECTION ACROSS X-AXIS
change to equation
change to coordinate
y = - f(x)
-y
SHIFT UP/DOWN
change to equation
change to coordinate
y = f(x) + k
y + k
(could also be subtraction)
HORIZONTAL STRETCH
change to equation
change to coordinate
y = f(x / h)
h * x
REFLECTION ACROSS Y-AXIS
change to equation
change to coordinate
y = f(- x)
-x
SHIFT RIGHT/LEFT
change to equation
change to coordinate
RIGHT:
y = f(x - k)
x + k
LEFT:
y = f(x + k)
x - k
