1EXAM- HPreCalc Flashcards
3 key numbers means
4 test regions
is (x-1)(x^2+x+1) a product of linear factors
no, simplify the last trinomial by doing quadratic formula to get a product of linear factors
do you put brackets around solutions that make the bottom 0
no
is x^2 + 49 factorable?
yes
x-7i) (x+7i
The theorem that can be used to determine the possible numbers of positive real zeros and negative real zeros of a function is called __________ __________ of ________
Descartes’s Rule, Signs
the formula that relates cost, revenue, and profit is ______
profit=revenue-cost
key numbers are ____
whatever makes the top and bottom 0
how do you find HA if the numerator’s degree=denominator’s degree
y=ratio of leading coefficients
f(x)=2x-5/4-x HA:y=-2
The _______ ________ _______ states that if f(x) is a polynomial of degree n (n>0), then f(x) has precisely n linear factors
linear factorization theorem
what is intermediate form? is it the same as undefined
0/0, no
The _______ _______ of _______ states that if f(x) is a polynomial of degree n (n>0), then f has at least one zero in the complex number system
fundamental, theorem, algebra
All graphs _____ the asymptotes
Follow
Increasing and decreasing intervals use the ___ values
X (watch asymptotes)
what is upper bound
you plug in a higher number and get all positive sums in return
this will be your highest zero
Make sure graphs don’t cross if there aren’t enough X intercepts
..
When finding key points of rational functions ___ first
Simplify
how do you find the vertical asymptote
whatever makes the denominator 0
roots of denominator
x=
be sure to factor function first as asymptotes might cancel
for the rational function f(x)=N(x)/D(x), if the degree of N(x) is exactly one more than the degree of D(x), then the graph of f has a ______ (or oblique) ________
slant asymptote
how do you factor a perfect cube
in the first (), take the cubed root of both variables in the equation
in the second (), first, put in whatever you need to get the leading coefficient, then put in the opposite of the product of the first (), then whatever you need to get the constant
slant asymptotes and parabolic asymptotes all begin with
y=
so do HA
VA begins with x=
If a+bi is a complex zero of a polynomial with real coefficients, then so is its ________, a-bi
conjugate
write x^6-x^7 as a product of linear factors
(x)(x)(x)(x)(x)(x)(1-x)
Why is x^4 + x^2-60 guaranteed two real roots
It’s down 60 going up eternally
Increasing and decreasing intervals use the ___ values
X (watch asymptotes)
A real number b is a ______ bound for the real zeros of f when no real zeros are less than b, and is a ______ bound when no real zeros are greater than b
lower, upper
To solve a polynomial inequality, find the _______numbers of the polynomial, and use these numbers to create __________ ________ for the inequality
critical/key, test intervals
what is DesCartes’s Rule of signs
the number of positive real zeros=the number of sign variations or less than that by an even integer
the number of negative real zeros=the number of sign variations at
f(-x) or less than that by an even integer
when f(x)—>+/- infinity as x—>a from the left or right, x=a is a _______ ________ of the graph of f
vertical asymptote
what does lower bound say
you plug in a negative number and get alternating sums in return,
this will be your lowest zero
Can an answer written as the product of linear factors have imaginary factors?
Yes, they are still linear
what does irreducible over the reals mean
does not reduce into real factors
x^2-2x+10
The test that gives a list of the possible rational zeros of a polynomial function is the ______ _______ Test
rational zero
in an inequality coordinate plane graph, the shaded area are the ______ and the line indicates ________
values that make the equation true, value that makes the equation =
what are the possible root combinations for a cubed equation
all real, 2 imaginary and one real
factor x^3+8
(x+2)(x^2-2x+4)
Can you cross through a HA? VA? SA?
Yes no yes
if 5i is a zero, what else is true
-5i is a zero
when you divide by a - in an inequality, ____ the sign
flip
key values with solutions in the middle are called ____ points
border
What is root 3i times root 3i
3i^2
-3
special cases are also called ______
unusual solution sets
what is the radicand
the polynomial under the root
A quadratic factor that cannot be factored further as a product of linear factors containing real numbers is said to be _______ over the _______
irreducible, reals
the key numbers of a rational expression are its ______ and its ___ ____
zeros, undefined values
what is a rational function
fraction with polynomials
what is the rational zero test
p(constant)/q (leading coefficient
some factors of both of these things will present a zero of the equation
Every polynomial of degree n>0 with real coefficients can be written as the product of _______ and ______ factors with real coefficients, where the ___________ factors have no real zeros
polynomial, linear, quadratic
how can you tell if an equation has 4 test regions
find solutions and make sure there are 3 roots
how do you find the horizontal asymptote if numerator’s degree is < denominator’s degree
y=0
it is the x-axis
a set of ordered pairs
relation
between two consecutive zeros, a polynomial must be entirely ____ or entirely _____
positive, negative
are solutions in interval notation ordered pairs
no
how do you find the HA if you numerator’s degree > denominator’s degree by 1?
by 2?
by 1: y=quotient slant asymptote
by 2: y=quotient parabolic asymptote
when f(x)—>b as x—>+/- infinity, y=b is a ______ ______ of the graph of f
horizontal asymptote
how do you find x intercepts and y intercepts of a rational function
x: roots of numerator (set=0) make sure to factor first in case things cancel
y: plug 0 in for x
NOTE- be sure to write as ordered pairs, (0,0) is not a y or x intercept
functions of the form f(x)=N(x)/D(x), where N(x) and D(x) are polynomials and D(x) is not the zero polynomial are called ______ _______
rational functions
the _______ is divided by the _______ which gives you a _____ with a ________
dividend, divisor, quotient, remainder
describe the behavior of the following graph
f(x)=7x^2-x^7
left up, right down (be careful with highest degree)
how do you find roots of f(x)=x^3-x^2+2
factors of p/ factors of q
+1, -1, +2, -2/+1, -1
test -1,1,2
The graphs of all polynomial functions are _______, which means that the graphs have no breaks, holes, or gaps
continuous
A ______ function is a second degree polynomial function, and its graph is called _________
quadratic, parabola
The _______ _________ Theorem states that if f is a polynomial function such that f(a) does not = f(b), then, in the interval [a,b], f takes on every value between f(a) and f(b)
Intermediate Value
when the graph of a quadratic function opens upward, its leading coefficient is ______ and the vertex is a _______
positive, minimum
what does upper bound mean
when you plug in a number into synthetic division and you get all positive numbers as a result, that will be your highest root
The ________ __________ _____ is used to determine the left-hand and right-hand behavior of the graph of a polynomial function
leading coefficient test
x^2+1 has ____ real solutions
no
what is standard form for complex numbers
a+bi
Critical points include
Relative max and mins and intercepts
imaginary roots come in _______, meaning a cubic function will have 2 roots with possible combinations ____…….
pairs,
3 real, 0 imaginary
1 real, 2 imaginary
NEVER 2 real, 1 imaginary
what is a rational number?
irrational?
- fraction using integers (4/7, 10) –decimal will repeat or terminate
- decimal that will never repeat nor terminate (pi, root2)
a set of ordered pairs
relation
The imaginary unit i is defined as i=_________, where i^2=___________
square root -1, -1
a+bi
if a=0, what number do you have?
pure imaginary
what is the pattern for imaginary numbers starting with i^1 and ending with I^4, before repeating itself over and over….
i (or, root -1), -1, -i, 1
what is the complex conjugate of root 6
root 6
The numbers a + bi and a-bi are called _________ ________ and their product is a real number a^2 + b^2
complex conjugates
The graph of a quadratic function is symmetric about its ____________________.
axis of symmetry
zeros of a polynomial are also called …..
solutions, factors, x-intercepts, roots
In the Division Algorithm, the rational expression f(x)/d(x) is _____ because the degree of f(x) is greater than or equal to the degree of d(x)
improper
how does a quartic graph behave
both sides go up/down
what is the standard quadratic function
f(x)=ax^2+bx+c
a cannot =0, c y intercept
an alternative method to long division of polynomials is calles _______ ______, in which the divisor must be of the form x-k
synthetic division (only when divisor is linear)
how do you rationalize 3+2i/6-i
multiply top and bottom by 6+i
A polynomial function is written in _______ form when its terms are written in descending order of exponents from left to right
standard
what is the position function (meters)
s(t)=-4.9t^2+Vot+So
all polynomial functions are ____ and _____
continuous, curvy
A _____ _______ number has the form a+bi, where a=0 and b does not equal 0
pure imaginary
When the graph of a quadratic function opens downward, its leading coefficient is _________ and the vertex of the graph is a _________
negative, maximum
you can only use synthetic division with __ functions
linear
A polynomial function of x with degree n, must have a __________ ___________ degree n and a _______ x
positive integer, real
what is the standard form for the equation of a parabola
f(x)=a(x-h)^2+k
a cannot =0
vertex (h,k)
When a is a positive real number, the _____ _____ root of -a ois defined as square root -a=square root ai
principal square
how many points of inflection can an equation have
degree-2
if f(a) (y value) <0 and f(b) >0 then in between a and b exists an _____
x-intercept
how do you find the vertex of an equation
-b/2a
plug result in for y part
The ______ Theorem states that a polynomial f(x) has a factor (x-k) iff f(k)=0
Factor
A factor (x-a)^k, k>1, yields a _______ _________ x=a of ______ k
repeated zero, multiplicity
how would you do
i^44+i^150-i^74-i^109+i^61
divide each power by 4, is remainder is 1, result is i, if the remainder is 2, result is -1, if remainder is 3, result is -i, if remainder is 0, result is 1
when writing complex numbers in standard form, what do you ALWAYS do first
convert to imaginary
ex) root -3 times root -12
=root 3i times root 12i
=-6
what is the conjugate of i
-i
describe the graph of g(x)=(3x)^2 +1
horizontal shrink, up 1
dividend=quotient*divisor+remainder
division algorithm
Intermediate value theorem is an ______ theorem
Existence
In the Division Algorithm, the rational expression r(x)/d(x)is _____ because the degree of r(x) is less than the degree of d(x)
proper
what 3 solutions do you test first with synthetic division
-1,1,2
When a real zero of a polynomial function is of even multiplicity, the graph of f _________ the x-axis at x=a, and when it is of odd multiplicity, the graph of f ______ the x-axis at x=a
bounces (touches) crosses
what are points of inflection
where concavity changes
What are extrema and extremum
Relative max and mins
True or False, i is a variable
False- it is not a variable, but treat it like it is when performing operations
when you test a positive number and get all positives in quotient, you have a __________ ____________
Upper bound
The _____ Theorem states that if a polynomial f(x)is divided by x-k, then the remainder is r=f(k)
Remainder
A _________ number has the form a+bi, where a does not equal 0 and b=0
real
how do you complete the square
make sure function is in standard quadratic form
group variable terms
make a=1 or factor out a
add (1/2b)^2
balance function (subtract ^ on outside)
factor trinomial (perfect square trinomial=2 same binomials)
rewrite in standard for for equation of parabola
REMEMBER–WHEN BALANCING, IF YOU FACTORED OUT AN A BEFOREHAND, CONSIDER THE ACTUAL VALUE TO SUBTRACT ON THE OUTSIDE
equation of a parabola= f(x)=2(x+3/4)^2 -65/8 what is vertex? axis of symmetry? x intercepts? y intercepts? image to y-intercept?
(-3/4, -65/8)
x=-3/4 (remember to include x=)
x intercepts- use quadratic formula or plug in 0 for y (original equation)
y intercepts- plug in 0 for x (original equation)
imagine- (-3/2, -7) (same y value, cross over axis)
dividend/divisor=quotient+remainder/divisor
alternative (division) algorithm
intermediate value theorem is also called ____________ ______________
existence theorem
what is the position function? (feet)
s(t)=-16t^2+Vot+So
- 16 is constant
- V is initial velocity
- S is initial position in feet
- t is time
every complex number has a ______ number (a) and an _____ number (b)
real, imaginary
Linear, constant, and squaring functions are examples of ____ functions
polynomial
A polynomial function of degree n has at most ___real zeroes and at most _____ turning points
n, n-1
are absolute value functions polynomials?
no- they are not curvy
what is the complex conjugate of (3+2i)
(3-2i)
- same exact term with a sign change in the middle*
- things will cancel*
When x=a is a zero of a polynomial function f, the following three statements are true:
a) x=a is a ______ of the polynomial equation f(x)=0
b) _____ is a factor of the polynomial f(x)
c) (a,0) is an _____ of the graph of f
zero, (x-a), x-intercept
A(n) ______ number has the form a+bi, where a does not equal 0 and b does not equal 0
imaginary
quadratic functions-
degree 2 polynomial
graph: parabola
cubic/linear is an example of a _________ rational expression
improper
when writing an imaginary quotient in standard form, what cannot be in the denominator
i, rationalize it
how would you do
2/1+i)-(3/1-i
do (2[1-i])-(3[1+i])
_______________
(1+i)(1-i)
what are the logarithmic models?
y=a+bLnx
y=a+blogx
what would you do with
y=2^-x^2
take the square of the number first then multiply by the implied negative one
the exponential function f(x)=e^x is called the ______ _____ function, and the base e is called the ______ base
natural exponential, natural
what is the product property of logs
goes to addition
log4X*y^2=log4X+log4Y^2
half life equations are decay, meaning k is
negative
logarithmic graphs have a ____ asymptote (__=#) while exponential graphs of a ____ asymptote (__=#)
vertical x
horizontal y
while doing the change-of-base formula, do you use the natural log or the common log
it does not matter as long as you are consistent
you can use the _______ Property to solve simple exponential equations
one-to-one
like terms have the same ____ and same _________
base, exponent
the domain of the natural logarithmic function is the set of __________ ________ __________.
positive real numbers
all real numbers such that x is greater than 0
is e a variable?
no, it is a constant
what is the equation for exponential growth?
decay?
what does each variable stand for
A=Ie^kt and A=le^-kt A is what you have (dependent) I is your initial amount e is 2.71828 (constant/base/multiplier) k is constant of exponentialism (constant coefficient/growth rate) and t is time (independent)
to evaluate a logarithm to any base, use the ______ formula
change-of-base
the inverse properties of logarithms state that logaA^x=x and
a^logaX=X
polynomial functions are examples of ___ functions
algebraic
the logarithmic function f(x)+lnx is called the ______ logarithmic function and has base _________
natural, e
how do you solve exponential or logarithmic (or any, really) equation
use inverse operations
logarithmic and exponential equations are _______
inverses
a logistic growth model has the form
y=a/1+be^-rx
if you have a log that can be taken by reducing the number, what do you do
reduce it, multiply it by the number you reduced it by, separate by addition, and finish
when solving an equation, it is important to check for _____________ by plugging your solutions back into the original equation
extraneous
to find the amount A in an account after t years with principal P and an annual interest rate r compounded continuously, you can use the formula _____________.
A=Pe^rt
in probability and statistics, Gaussian models commonly represent populations that are ___________ _____________.
normally distributed
how would you simplify 3^x-2
3^x3^-2
3^x(1/9)
the inverse function of the exponential function f(x)=a^x is called the ______ function with base a
logarithmic
for like terms with the same base and exponent, when multiplying, add exponents but ______________________ the bases
do not do anything with the bases
a logarithm is an _____
exponent
how you know if an equation is exponential
it has a variable as an exponent
what is the quotient property of logs
to difference
log7 x^3/y=log7X^3-log7Y
how can you make powers roots and roots powers
the cubed root equals ^1/3 and so on
what does e=
2.71821
when simplifying logs, make sure to use parentheses between subtraction and addition and also make sure that the log of a certain number–make sure that number cannot be divided by anything to get a whole number answer. if it can, multiply and add based on rules of expansion
..
what is the power property of logs
logx^4=4*logx
(make sure you only bring it out if it is for the WHOLE THING
exponential and logarithmic functions are examples of nonalgebraic functions, also called _____ functions
transcendental
your equation is done when _ is by itself
x
when solving an exponential equation, take the ___ of both sides
log
to one-to-one property of natural logarithms states that if Inx=Iny, then_______________
x=y
asymptotes begin with _________________________.
x= or y=
An exponential growth model has the form ______, and an exponential decay model has the form_________.
y=ae^bx or A=Ie^kt
y=ae^-bx or A=Ie^-kt
remember to use ________ when you are solving logs to indicate separation
parentheses
to find the amount A in an account after t years with principal P and an annual interest rate r compounded n times per year, you can use the formula______________.
A=P(1+r/n)^nt
what is true of all logarithmic graphs?
how do you restrict the domain?
how do they look?
they all pass vertical line test, always have x-intercept and asymptote
you cannot take the log of 0 or a negative number
boomerang
you can consider logaX to be a constant multiple of logbX; the constant multiplier is ______________
1/logbA
the parent log has an assumed base of ____
the natural log has an assumed base of __
10 (common log)
e
what is the log of 625 to base 5
what is the log of .001
4, -4
when you are told to find an exponential model, make sure you have all variables expect____
the x and y, or the A and t
the common logarithmic function has base
10
the change-of-base formula for base a is given by logaX=__________
logbX/logbA
describe the graph
h(x)=(x+2)^3 +1
cubic function shifted two units to the left and shifted up one
what is (f-g)(0)
f(0)-g(0)
when is there a reflection in the y-axis for a function?
x-axis?
y-axis: h(x)=f(-x)
x-axis: h(x)=-f(x)
for a function to have an inverse, it must pass (Horizontal/Vertical) Line Test
BOTH- it must pass vertical to be a legitimate function. It must pass horizontal to have an inverse
The graphs of f and f^-1 are reflections of each other in the line ___
y=x
what is f^-1(x)
inverse of f (-1 has new mathematical value!)
g(x)=(x-1)^3 +2
what is the parent function?
use function notation to write g in terms of f
f(x)=x^3
g(x)=f(x-1) +2
***Remember not to include the ^3, as that is implied in f(x)
an inverse cannot fail (Horizontal/Vertical) line test
I think just Vertical???
The domain of a function = the _____ of its inverse.
the range of a function = the _____ of its inverse
range, domain
The _____ of the function f with g is (f of g)(x)=f(g(x))
composition
when will a function vertically stretch?
Shrink?
n*f(x) when n<-1(-infinty,-1) or n>1(1, infinity)
n*f(x) when -1<n<1 (-1,1)
f(1)=4 really is
an ordered pair (1,4)
f(x)=x^2 +6 g(x) square root (1-x)
divide these
x^2 +6/square root of (1-x) —cannot have square roots on the bottom (multiply top and bottom by square root (1-x))—-
x^2 +6 square root (1-x)/1-x
The linear model with the least sum of square differences is called the ______ ______ _______ line
line of regression
how do you decompose a composite function? decompose h(x)=1/(x-2)^2
first find the simplified function (this is your f(x)), then think, what do i plug in (g(x)), to get what I have now?
f(x)=1/x^2 g(x)=x-2
***Note that the ^2 is outside of the parentheses. This is why you cannot have f(x)=1/x and g(x)=x-2^2–> you could have g(x)=(x-2)^2
how would you use variation terminology to say A=1/2bh
the area of a triangle is jointly proportional to its base and height
what must you have in order to find k-the constant of variation
initial condition
what does a graph’s inverse do
what kind of symmetry do they have
switches x and y
reflectional symmetry over the line y=x
how do you prove f(x) and g(x) are inverses of eachother
use composite functions (analytically) plug g(x) into f(x) [f(g(x))] and you will get x
what is the composition of functions
taking one function and plugging it into another function (not commutative)
If the composite functions f(g(x)) and g(f(x)) both equal x, then the function g is the ______ function of f
inverse
when performing a piecewise function, always ______ your solutions and make sure that your functions work
test
An r value of a set of data, also called a ________ _________, gives a measure of how well a model fits a set of data.
what is the worst of these? best?
correlation coefficient
0, 1
The inverse function of f is denoted by
f^-1
a set ordered pair (mapping, x/y chart, etc.)
relation
transformations that cause shapes to change (horizontal or vertical stretches) are _____________
example?
nonrigid transformations
dilation
is composition of functions commutative? what does this mean?
no- this means that you will get the same answer in reverse (think addition, multiplication [4+2=2+4])
what is the equation for state income tax
what kind of variation does it have
state income tax=k(gross income) (T=k*g where T is the dependent variable and g is independent)
direct
a constant function can be horizontal and vertical
true or false
false–ONLY HORIZONTAL
y varies directly as x
y is directly proportional to x
y=kx for some nonzero constant k
direct variation
Statisticians use a measure called the ______ of _____ _______to find a model that approximates a set of data most accurately
sum, square differences
Two techniques for fitting models to data are called direct and iverse _______ and least squares ________
variation, regression
A nonrigid transformation of y=f(x) represented by g(x)=cf(x) is a _________ when c<-1 or c>1 and a ______ _______ when -1<c<1
vertical stretch, vertical shrink
what are the 4 types of functions
polynomial, rational, radical, trigonometric (i dont think we need to know this just in case though)
what is a defined function
one that has a domain of all real numbers
The domain of f is the ____ of f^-1 and the, and the ______ of f^-1 is the range of f
range, domain
The joint variation model z=kxy can be described as “z varies jointly as x and y,” or “z is ________ ________ to x and y.”
directly proportional
a piecewise defined function will always have at least one x-intercept or at least one y-intercept
true or false
true- defined means that the domain is all real numbers so it will have a y intercept at least
is (f of g)(x) muliplication
No
If f(x)=x^2 and you plug in 4t, what do you get
(4t)^2=16t
In direct variation models of the form y=kx, k is called the ____ of ____
constant, variation
the constant function (f(x)=c) and the identity function (f(x)=x) are two special types of _____ functions
linear
describe the translation f(x)+c f(x)-c f(x+c) f(x-c)
c units up
c units down
c units left
c units right
when the shapes are congruent or unchanged after a transformation, the transformation is a _________
what kinds of transformations are included
rigid transformation
translations, reflections
y=k/x is ______ variation. It is the opposite of _____
inverse, direct
what is mathematic modeling
coming up with the equation
how would you make f(x)=x^2 a function with an inverse
restrict the domain (x>_ 0)
a function is __________ if it passes Horizontal and Vertical line test
one to one
only __________ have an inverse function
one to one
describe the graph
j(x)=-(x+3)^2 +1
quadratic function reflected over the x-axis, shifted three units to the left and up one
what is k in y=kx?
constant of variation, also the rate
Inverse operations _________ each other
undo
in (f of g)(x), what is the domain?
the domain of f of g is the set of all x in the domain of g such that g(x) is in the domain of f
A reflection in the x-axis of y=f(x) is represented by h(x)= ________, while a reflection in the y-axis of y=f(x) is represented by h(x)=__________
-f(x)
f(-x)
what are the three types of transformations
what are the four types of translations
translation, reflection, dilation
up, down, left, right
f(x)=square root x
parent radical function
two functions f and g can be combined by the arithmetic operations of ________,_________,_________, and ____________to create new functions
addition, subtraction, multiplication, division
S=4pi r^2
how would you use variation terminology to say this aloud
The surface area of a sphere varies directly as the square of the radius r (your constant of variation is 4pi [you will never say a number])
what is the equation for Interest
I=Prt
since k is also rate, you could say I=k(P)(t
To reflect over x axis, make _ values negative
Vice versa
y
horizontal shifts, vertical shifts, and reflections are called _____ transformations
rigid
Be careful with distributing negatives in reflection cases
Square root of (x+6) reflected in both x and y axes is…
-square root of -x-6
what kind of variation will these ordered pairs have?
5, -3.5)(10, -7)(15, -10.5)(20, -14)(25, -17.5
Direct—although technically the numbers are getting smaller, the positive values are increasing
f(x)=x^2
parent quadratic function
f(x)=1/x
parent rational function (reciprocal function)
in a piecewise function, the ranges of the starting functions are the _________ of the inverse functions
domains
The mathematical model y=k/x is an example of _____ variation
inverse
Remember when a graph is up 3 and a point is (1,7), it’s technically
(1,4)
how do you find a lines equation and graph a scatter plot on your calculator?
lines equation given a lot of points- hit stat, edit, enter x values in L1 and y values in L2, make sure you have the same number of data entries, hit stat again, calc, LinReg, make sure Xlist says L1 and Ylist says L2, calculate
this will give you your equation!!!
to plot a scatter graph- go to y=, graph the line you got above^ (may have to adjust windows) then hit Stat Plot (second y=), plot 1 on, choose type, graph (should get a line with plots)
inverse variation says as one gets bigger, _________________________
the other gets smaller
what are transcendental functions
mix of two types of functions (1/x^2) quadratic and rational
A function is _____ when each value of the dependent variable corresponds to exactly one value of the independent variable
one-to-one
the greatest integer function takes the next integer ______
so -3.1 would go to __ and 2.9 would go to ___
-4, 2
it is also called the round-down function by some
direct variation models can be described as “y varies directly as x,” or “y is _______ ________ to x”
directly proportional
f(x)= [[x]]
greatest integer function (or, step function)
f(x)=x
identity function
a linear equation will always have an x intercept and a y intercept
true or false
false-constant functions will not have an x intercept
A graphical test for the existence of an inverse function of f is called the _____ Line Test
Horizontal
z=kxy
joint variation (z varies jointly as x and y)
what is a piecewise function
a function with pieces (normally 2 or 3)
The direction variation model y=kx^n can be described as “y varies directly as the nth power of x,” or “y is ____ _____ to the nth power of x”
directly proportional
what is the difference between 1/x and x/1 in terms of functions
1/x is the parent rational function and x/1 is a linear function with slope 1/1
z varies directly with the square of x and inversely with y with a constant variation of 2/3
how would you write this?
z=2x^2/3y (separate your fraction)
what is a model?
an equation
what would the graph x=2 look like?
y=3?
vertical line through 2
horizontal through 3
what is a piecewise function
a function with multiple equations, each with designated rules
In calculus, one of the basic definitions is that of a ___________, given f(x+h)-f(x)/h, h cannot =0
difference quotient
true or false– if an equation has a y value which produces 2 x values, it is NOT a function
false- although an x value cannot produce two or more y values, a y value can produce two or more x values
The _______ of a function f are the values of x for which f(x)=0
zeroes (roots, x-intercepts)
The ______ ______ _______ _______ between any two points (x1, f(x1)) and (x2, f(x2)) is the slope of the line through the two points, and this lines is called the _____ line
average rate of change, secant
what are points of inflection
where concavity changes
translation means ____
slide
what axis does your independent variable (x) go on? (y)?
x, y
what is slope intercept form
y=mx+b
when you construct and use a table to solve a problem, you are using a ______ approach
graphic
what is a function
a specific relation saying your x value can never produce two or more y values
what is a secant line
a line that intersects two points
When the x-axis and the y-axis have different units of measure, the slope can be interpreted as a ______
rate
which way do you view a graph to tell if it is increasing or decreasing
left to right
An ordered pair of real numbers can be represented in a plane called the rectangular coordinate system or the ____ plane
cartesian
The ________ ________ ________ is used to determine whether the graph of an equation is a function of y in terms of x
Vertical Line Test
The set of all solution points of an equation is the ___ of the equation
graph
how would you find the equation of a line parallel to y=3x-7 through (4,1)
your slope is 3, then plug in 4 for x, 1 for y, and 3 for m (in slope intercept) and solve for b.
OR–plug (4,1) in point slope form with m as 3
If the domain of the function f is not given, then the set of values of the independent variable for which the expression is defined is called the ___________.
implied domain
what is the equation of a circle
(x-h)^2+(y-k)^2=r^2
center (h,k)
radius r
when writing an equation as a function of another equation, look for
shared variables
A function f is ______ on an interval when, for any x1 and x2 in the interval, x1 is less than x2 implies f(x1) is greater than f(x2)
decreasing
how do you know if an equation’s graph has x-axis symm? y-axis? origin?
x- plug in negative for y, must be the same
y- plug in negative for x, must be the same
origin- plug in - for both, must be the same
how do you find average rate of change
F(new x) - F(old x)/new x-old x
the ___________ is a result derived from the pythagorean theorem
distance formula
height=
length=
top-bottom
right-left
if the graph of a function was a) even or b) odd. what ordered pair could also lie on that graph if (-5/3, -7) did
a) (5/3, -7)
b) (5/3, 7)
The points at which a graph intersects or touches an axis are called the ______ of the graph
intercepts
distance formula is derived from ______
pythagorean theorem
what does { mean?
: ?
set, such that
Two lines are ______ iff their slopes are negative reciprocals of each other
perpendicular
what is percentage increase (or decrease)
amount increase/original amount
an odd exponent (x^3) signifies what
that the graph will got through each intercept (no repeats)
A graph is symmetric with respect to the _____if, whenever (x,y) is on the graph, (-x,y) is also on the graph
y-axis
For an equation that represents y as a function of x, the set of all values taken on by the ________ variable x is the domain, and the set of all values taken on by the ______ variable is the range
independent, dependent
how do you solve a difference quotient
make two large brackets, plug in when they are asking you to. simplify and solve
The prediction method __________ is the method used to estimate a point on a line when the point does not lie between the given points
linear extrapolation
if your function is not odd or even, what is it? what kind of symmetry does this kind of function have
neither–no symmetry
What is point slope form
y2-y1=m(x2-x1)
what is domain
set of all values that the independent variable (usually x) can be
what does an open point signify on a graph
that breaks one of the rules in your piecewise function rules
how would you express that your domain can be all real numbers greater than -2 using interval notation
Domain= {R’s: (-2, [infinity sign])}
Two lines are ______ iff their slopes are equal
parallel
when a graph is constant, you always express the ordered pair with _______
brackets [ ]
Every line has an equation that can be written in __ form
standard
intercepts are written in
ordered pairs
true or false- a function with a square root cannot have a domain that is the set of all real numbers
false– the x value could be raised to an even root underneath the square root
i.e. y=square root of x^2
Polynomial functions are _________ and ___________
continuous and curvy
how do you express that the domain of a function is x greater than or equal to 1 in interval notation. What notation is it in above?
[-1, infinity) inequality
Finding the average values of the representative coordinates of the two endpoints of a line segment in a coordinate plane is also known as using the
midpoint formula
A relation that assigns to each element x from a set of inputs, or _________, exactly one element y in a set of outputs, or ______, is called a ______.
domain, range, function
An ordered pair (a,b) is a ______ of an equation in x and y when the substitutions x=a and y=b result in a true statement
solution
The point of intersection of the x axis and the y axis is the ________, and the two axes divide the coordinate plane into four parts called ____
origin, quadrants
what is implied domain
values acceptable for x for a certain function (given)
usually in in piecewise
what does relation mean
set of ordered pairs (mapping, x/y chart, etc.)
what is the variable under the radical called
radicand
what is the range
the set of all values that the dependent variable (usually y) can be
the simplest mathematical model for relating two variables is the ____ equation in two variables y=mx+b
linear
what is (x^2-1) said to be
difference of perfect squares
what is intercept form
x/a + y/b = 1
when a does not equal nor b does not equal 0
what is another name for the coordinate system, where does it come from
cartesian plane, Rene Descarte
what is slope formula
y2-y1/x2-x1
A function f is _____ when, for each x in the domain of f, f(-x)=-f(x)
odd
The _____________ form of the equation of a line with slope m passing through (x1, y1) is y1-y2=m(x2-x1)
point slope
what are the types of transitions
dilation, reflection, rotation
what is f(1)=4 in simpler terms?
(1,4) an ordered pair
What is standard form
Ax+By=C
what is the parent function for absolute value?
y=IxI
How can you tell if a function is even? What kind of symmetry does it have?
Plug in -x for x, and your equation will remain exactly the same {{f(x)}}
Reflection symmetry over the y-axis
For a line, the ratio of the change in y to the change in x is called the __ of the line
slope
how do you know if a function is odd? What kind of symmetry does it have?
Plug in -x for x and your equation will be -f(x) (exact opposite). It has rotational symmetry 180 degrees about the origin
what is the difference between intercepts, roots, and zeroes?
no difference
how do you determine if an equation represents y as a function of x
set it equal to y–determine if one x value could produce two y values
A ________________ is a location on a graph where your line stops increasing and starts decreasing
relative max
