U1 A1- Polynomial Functions And Operations Flashcards
Do you factor the simplified expression once you add/subtract polynomial functions
No
In f(x)-g(x) what do you have to remember
Put ALL of g(x) into brackets and apply the negative to the whole expression
What is the order of operations for -x^2
1) square first
2) then apply the negative sign
What does it mean when a function is defined
It has an equation
When functions aren’t defined how do you find the sum or difference
Add/subtract the y values (outputs) at a given x value (input)
Given:
F(-3)=5
g(-3)=8
What is f(-3) + g(-3)
13
The domain of the combined sum or difference function may be ______ than the _____ domain
The range of the combined function can be determined from the _____
Different
Individual
Graph
F(x) domain: (-infinite,3)
g(x) domain: (-5,infinite)
What is the domain of (f+g)(x)
(-5,3)
The domain and range of a graph is always ____ unless there are restrictions
Infinite
x is an element of real numbers
How do you know if domain and range aren’t infinite when just given equation
If there are restrictions in the equation
What are 2 examples of restrictions on an equation
- x under a square root sign
- dividing something by x
What is the constant of a quadratic function (in terms of graphing)
y-intercept
The domain of the product or quotient function may be ______ than the _____
Different
Individual
The range of a combined function (multiplication or division) can be determined from the ____
Graph
How do you draw a new graph when you combine two functions
1) make a table and state the y-value of f(x) and g(x) at x values that you pick
2) combine the two y-values however you’re supposed to (add/sub/mul/div) And those are your new y-values
When you are determined a simplified expression for multiplication or division of polynomials should you factor after
No
How to determine the degree of a function multiplication statement without foiling it out
Count the x’s
(Linear function )(linear function)= _________
and why
Quadratic
(x)(x)=x^2
What is always the domain of a quadratic unless there are restrictions
Infinite
x is an element of real numbers
When is the only time you can add/subtract radicals
When the radicand are like terms
What is a radicand
Stuff under root sign
Before you state NPV’s what do you always have to do
Factor the expression
A discontinuity is a _____ in the graph
Graph
Vertical asymptote- a line that the graph _____ but never touches
Approaches
A vertical asymptote occurs when a ration expression cannot be ______ further
Simplified
Identify the vertical asymptote in
y=(x-4)/(x-1)
Restriction: x can’t equal (1)
Since ORIGINAL EXPRESSION can’t simplify further,
x= (-1) is vertical asymptote
Point of discontinuity- a single ___ that doesn’t exist on the graph
Point
A point of discontinuity occurs when a rational expression ___ be simplified further
Can
Identify the point of discontinuity in the expression
y= [(x-1)(x+2)]/(x-1)
Restriction: x can’t equal (1)
Expression can be simplified further to (x+2) so x=1 is a point of discontinuity
Then sub the x value into the expression to get the y-value
The y-value of a point of discontinuity can be found by _______ the restricted x-value into the _____ expression
Substituting
Simplified
What is quotient to anything divided by 0
And what does this mean in terms of functions
Undefined
Know that this is a point of discontinuity
(look at the x-value and plug it in to find the y value of the hole)
(Quadratic function)/(linear function)=_______
and why
Linear function
X^2)/(x)=(x
How do you solve 3(fg)(2)
1) evaluate the product of (f)(g) when x is 2
2) multiply the whole thing by 3
When adding/subtracting polynomials fractions you need ______
LCD’s
Can you cancel binomials in division if they’re not the EXACT same expression
No
When multiplying/dividing functions a NPV will result in a ______ or _____
Vertical asymptote
Point of discontinuity
To draw the graph of functions being multiplied/divided you multiply the two ____ values at the same ____ value
Y
X
Composite functions- ____ within functions
Functions
How do you write (f dot g)(x) as a composite function
f[g(x)]
To find the equation for composite functions, ______ the inside function for x in the ____ function
Substitute
Outside
To evaluate composite functions, evaluate the ____ function at the given ___ value then take this output and evaluate the _____ function at the ____ x value
Inside
X
New
How to find domain of composite functions
Combine the domains of each function
Identify any restrictions and state domain (as we always do)
Composite functions- if the inner function isn’t defined, neither is the _____ function
Composite
What does a dot with no colour mean
A functions OF a function
How to determine f[g(4)] using a graph
1) Find y value of g(x) graph at (4)
2) find y value of f)x) graph at whatever the new x value is
Polynomial functions are characterized by their ____, _____, and constant term
Degree
Leading coefficient
Graphs- odd degree polynomials have ____ arms
Opposite
Odd degree polynomials- if the leading coefficient is GREATER than 0, the graph will ____ from left to right
Rise
Odd degree polynomials- if the leading coefficient is LESS than 0, the graph will ____ from left to right
Fall
Graphs-even degree polynomials have arms going the ____ direction
Same
Are negatives whole numbers
Yes
Even degree polynomials- if the leading coefficient degree is GREATER than 0, the graph will open ____
Up
Even degree polynomials- if the leading coefficient degree is LESS than 0, the graph will open ____
Down
What does the constant term represent in terms of graphing
Y-intercept
All polynomial functions exhibit _______
Abs what does this mean
Continuity
No breaks in the graph, so no restrictions on the domain
Polynomial functions- all tents MUST have ____ number coefficients
Whole
Polynomial functions- the leading coefficient must be a ___ number
Real
What is a real number
Can be defined
How do you draw a graph for a function given the characteristics of the function
Look at degree, LC, and constant
Draw a graph based off of the identified characteristic in each of those
When drawing a graph if the constant term is negative what do you have to remember
The y-intercept has to be negative
Odd degree functions- arms go in _____ directions
Opposite
Even degree functions- arms go in ____ direction
Equal
Max # of turning points of a graph is less than the ______
Degree
What is the dividend
that is being divided
What is the divisor
that another # is being divided by
What is the non- simplified formula for division of polynomials
(dividend/divisor)=Q +(remainder/
divisor)
What is the simplified formula for division of polynomials
dividend= divisor(Q) + remainder
WHAT IS THE MOST IMPORTANT THING WITH LONG DIVISON OF POLYNOMIALS
(IF YOU DONT DO THIS YOUR ANSWER WILL BE WRONG)
Any powers of the variable that are missing must be included with a 0 coefficient
Ex) 3x^3 + 0x^2 + x +9
What is the remainder Therom
When P(x) is divided by a binomial, the remainder can be determined by evaluating P(a), where a is the root of the divisor
What is the formula for remainder therom
And what do the variables represent
P(a)= R
a= root of the divisor
R=remainder
When question gives you dividend and divider and asks for remainder, should you use long divison
(And why)
NO
(Use remainder therom
Steps for using remainder theorem to find the remainder
1) find the root of the divisor
(Let equation equal 0 and solve)
2) plug that value (a) into dividend to solve for remainder
Factoring polynomials: if a number is a factor of another number, it can perfectly divide into the second number without a ____
Remainder
The factor theorem is an _____ of the remainder theorem
Extension
The factor theorem: when a polynomial is divided by a binomial such that the _____ is 0, then the binomial ____ a factor of the polynomial
Remainder
Is
Factoring polynomials: the root of the divisor is a ___ or ___ ____ of the polynomial
(Means it’s both)
Zero
X-intercept
Factoring polynomials: what does a 0 output mean
You have a perfect root, so a perfect factor
Steps for factoring polynomials degree 3 or higher
1) list all the factors of the constant term (pos and neg)
2) evaluate the expression at each of the factors until one of them =0
(Sub number in for all the x values)
3) find the factor from that since
x= whatever number was correct
4) divide expression by that factor
5)
- if quotient is degree two use decomp to find the other two factors
- if quotient is degree three or higher repeat the same process again (keeping in mind that your final answer will have that first factor you divided by in it)
How to find the roots, given the factors
let each factor equal 0 and solve for x
How are the quadrants labelled on a graph
Top right=1
Top left=2
What is the min and max number of zeros a degree 5 polynomial can have
(And why)
1-5
- odd degree means it has opposite arms (so has to cross x axis at least once)
- 5 since it’s degree five
What is the min and max number of zeros a degree 4 polynomial can have
(And why)
- even degree means it’s arms go in the same direction (so doesn’t have to cross x axis even once)
- 4 since it’s a degree 4
Multiplicity refers to how many times a particular ____ of a function ____
Zero
Repeats
WHAT 4 THINGS DO YOU HABE TO INDENTIFY IN AN EQUATION IN ORDER TO DRAW THE CORRESPONDING GRAPH
- multiplicity
- degree
- end behaviour
- constant
What is the end behaviour of a graph and how do you determine it
(Both odd and even degree polynomials)
If the graph falls or rises
Odd:
Negative lead coefficient= falls
Positive lead coefficient= rises
Even:
Negative lead coefficient= opens down
Postive lead coefficient=rises
What does a multiplicity of 2 mean
The factor repeats two times
Multiplicity: what is the behaviour if there’s a multiplicity of 1
Crosses x axis
At whatever x equals
Multiplicity: what is the behaviour if there’s a multiplicity an even #
Touches x axis
At whatever x equals
Multiplicity: what is the behaviour if there’s a multiplicity an odd #
Bends at x axis
At whatever x equals
Multiplicity: after you determined the behaviour, how do you know where the behaviour happens
Find the root of the factor
Multiplicity: how many different behaviours are there
3
How to determine a transformation equation from a point on the original graph
Pick a point on the original graph
1) determine stretch
2) apply transformation to point
3) determine reflection
4) apply transformation to point
5) determine translation
6) apply transformation to that point
WHAT CAN’T YOU DO WHEN DETERMINING A TRANSFORMATION FROM AN ORGINAL AND TRANSFORMED GRAPH
Pick a point on original and pick corresponding point on transformed and determine equation from that
How to solve:
For what value of b will
P(x)= 4x^3 -3x^2 + bx +6
Have the same remainder when it is divided by both (x-1) and (x+3)
1) Sub each root in the equation and simply
3) P(1)=P(-3) and solve for b
Steps for solving by substitution
1) simplify two equations (there should be at least one x and y in each)
2) manipulate one equation (doesn’t matter which), so that one variable is in terms of the other
3) solve that equation (now you have the value of one variable)
4) sub the value you just found into the other simplified equation (to determine the value of the other variable
If h(x)= f[g(x)] determine f(x) and g(x) when:
h(x)= 2x^2 -1
1) g(x) has to have an x in it in Oder to be a function so let it equal x^2
2) work backwards from there (you can see that x^2 is being subbed into 2x-1) so that is the function of f(x)
How to solve:
For what value of c will
P(x)= 2x^2+ cx
Have the same remainder when it’s divided by x-2 and x+1
1) find the two roots
2) make each expression equal each other and sub in the roots p(-1)=p(2)
3) simplify and solve for c
How to do this:
When 3x^2 + 6x -10 is divided by (x+k), the remainder is 14. What is the value(s) of k
1) start as usual with P(a)=R
2) make equation equal 0 after you’ve simplified and factor to find values of k
How to do this:
When the polynomial
mx^3 - 3x + nx +2 is divided by (x-2) the remainder is 1. When it’s divided by (x+3) the remainder is 4.
What are the values of m and n
1) sub the roots in and simplify the equation as much as you can
2) add the equations to find the value of one variable
3) sub that value onto one of the simplified equations to find the value of the other variable
How to do this:
If h(x)=f[g(x] and h(x)=(2x-5)^2 and f(x)= x^2 what is g(x)
And what’s the answer
Inside term is always going to be the inside function so g(x)=2x-5
And you can clearly see this makes sense if f(x)=x^2
When you’re dividing by two fractions how do you set it up in a way that you can easily solve
Multiply the top fraction by the reciprocal of the bottom fraction
How to find the new domain and range of a function after you add/subtract/divide/multiply
Graph it
What are 3things that make a polynomial function NOT a polynomial function
- x under the root sign
- fraction exponent
- negative exponent
If a number is divided just by x is it a polynomial function
No (it has a discontinuity)
If part of a polynomial function is -2x^2 with the -2 under the root sign is it a polynomial
No (anything squared is positive and positive times negative is negative, which you can have under a root sign)
What is a rational function
Both numerator and denominator are polynomials
What is a radical function
Has a square root with x in the radicand
When finding multiplicties if there’s an x by itself what do you have to do
Include a multiplicity with that as well
