Exam 3

Question 1

zero of a polynomial

Answer 1

number “k” such that f(k) = 0

if k is a real number, it is an x-intercept

Question 2

MUST be a factor of a polynomial

Answer 2

(x-k) where k is a zero of the polynomial

Question 3

synthetic division can be used when…

Answer 3

dividing by (x - k)

Question 4

used to get a factor from a polynomial that can’t be factored

Answer 4

rational zeros theorem

Question 5

define rational zeros theorem

Answer 5

for any polynomial with a leading coeffecient of Q and a constant of P, any rational zero must =

factor of P / factor of Q

Question 6

steps to rational zero theorem

Answer 6

1. find Q (leading coeffecient) and P (constant)
2. find factors of Q and P
3. set up all possible divisions of a factor of Q by a factor of P, and eliminate duplicates
4. plug into function until you find one that makes f(x) = 0
5. make a factor from this zero and divide from polynomial

Question 6

steps to rational zero theorem

Answer 6

1. find Q (leading coeffecient) and P (constant)
2. find factors of Q and P
3. set up all possible divisions of a factor of P by a factor of Q (p/q), and eliminate duplicates
4. plug into function until you find one that makes f(x) = 0
5. make a factor from this zero and divide from polynomial

Question 7

domain of a rational

Answer 7

set of all x-values such that the denominator =/= 0

Question 8

how to find LCD for polynomials

Answer 8

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

Question 9

what to check for after solving rational equations

Answer 9

must make sure x-value does not cause division by 0

Question 10

how to find x-intercepts in rationals

Answer 10

set equal to 0

disregard denominator

find zeros of numerator

Question 11

m. 1 effect on x-int

Answer 11

straight through

Question 12

m. even effect on x-int

Answer 12

bounces off

Question 13

m. odd (except 1) effect on x-int

Answer 13

S-shape

Question 14

m. odd effect on vertical asymptote

Answer 14

opposite directions

Question 15

m. even effect on vertical asymptotes

Answer 15

same direction (volcano or trench)

Question 16

how to find VA

Answer 16

find x-values that make denominator = 0

Question 17

horizontal asymptotes are part of ______ behavior

Answer 17

end

Question 18

vertical asymptotes are part of ______ behavior

Answer 18

middle

Question 19

equation for HA

Answer 19

y = #

Question 20

how to find HA

Answer 20

leading coeffecient of numerator / leading coefficient of denominator

if # / x, then y = 0

if x / #, it is a slant asymptote

Question 21

used to solve polynomial inequalities

Answer 21

analytical technique

Question 22

analytic technique steps

Answer 22

1. put 0 on one side of equation
2. ID x-ints and VAs
3. mark on a number line
4. test values in each interval on the number line in the function
5. solution is the union of the true intervals (VAs are never included, x-ints are if inequality includes =)

Question 23

A^b/c =

Answer 23

^c√A^b

denominator goes out; numerator stays in

Question 24

necessary after solving radical equations

Answer 24

check work - there may be no solution if the square root of a number gives you a negative value!

Question 25

define 1-to-1 function

Answer 25

function in which each y value connects to only one x value

Question 26

1-1 functions must pass…

Answer 26

both vertical & horizontal line tests

Question 27

if x leads to multiple y values…

Answer 27

not a function

Question 28

if y leads to multiple x values…

Answer 28

not a 1-1 function

Question 29

difference quotient =

Answer 29

f(x + h) - f(x) / h

Question 30

if all is correct in DQ, h will…

Answer 30

cancel

Question 31

2 exceptions to domain of all real numbers for functions

Answer 31

• asymptotes of rationals caused by division by 0
• square root functions, in which whatever is beneath the √ must be > or = 0

Question 32

“slope” of nonlinear functions =

Answer 32

average rate of change =

f(b) - f(a) / b - a

Question 33

define function extrema

Answer 33

minimum and maximum points on a graph

y-values only

infinity does not count

Question 34

define absolute minimum and maximum

Answer 34

highest and lowest y-value on a graph that is not infinity

Question 35

define local maximum and minimum

Answer 35

output of a function where the function switches from decreasing to increasing OR increasing to decreasing

switching to a constant interval does not count

Question 36

find domain of a composed function

Answer 36

1. find domain of the final composed function
2. unite it with the domain of the inner function

Question 37

find domain of a combined function

Answer 37

intersection of domains of original functions

Question 38

define transformation

Answer 38

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

Question 39

3 vertical (y-coordinate) transformations

Answer 39

vertical stretch/squish

reflection across the x-axis

shift up or down

Question 40

3 horizontal (x-coordinate) transformations

Answer 40

horizontal stretch/squish

reflection across y-axis

shift left or right

Question 41

VERTICAL STRETCH

change to equation

change to coordinate

Answer 41

y = f(x) * h

y * h

Question 42

REFLECTION ACROSS X-AXIS

change to equation

change to coordinate

Answer 42

y = - f(x)

-y

Question 43

SHIFT UP/DOWN

change to equation

change to coordinate

Answer 43

y = f(x) + k

y + k

(could also be subtraction)

Question 44

HORIZONTAL STRETCH

change to equation

change to coordinate

Answer 44

y = f(x / h)

h * x

Question 45

REFLECTION ACROSS Y-AXIS

change to equation

change to coordinate

Answer 45

y = f(- x)

-x

Question 46

SHIFT RIGHT/LEFT

change to equation

change to coordinate

Answer 46

RIGHT:

y = f(x - k)

x + k

LEFT:

y = f(x + k)

x - k