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Unit 4 - Coggle Diagram
Unit 4
4.1
Vertex
highest or lowest point on a graph
f(x)=a(x-h)2+k
Axis of symmetry
the line that splits the graph in two
x- intercept
points where the graph intersects the x-axis
Vertex of a parabola
(-b/2a,f(-b/2a))
Opening of a graph
a positive coefficient means the graph opens up
negative leading coefficient means that the graph opens down
4.2
Horizontal projectile motion
the length in which an object is launched
has a maximum value
Maximize functions
how much an item can be sold to maximize profit
Maximized profit
the greatest amount of profit made after a curtain amount of revenue has been used
read problem to make equation
4.3
Polynomial function
can be found by looking at the leading constant
Leading coefficient
if it is greater than zero the right hand behavior finishes up
if it less than zero then the right-hand behavior finishes down
Degree's on a graph
if a the degree of a graph is odd the the graph will start and finish in opposite directions
if the degree is even then the graph states and ends in the same direction
Shape of a polynomial functionary zero multiplicity of K
if k is greater than 1 then the graph is even and touches the x-axis
if k is greater than of equal to 1 then it is odd and crosses the x-axis
Sketching a polynomial function graph
find end behavior
plot the y-intercept
find all all really zeros and there multiplicities
find test value & sketch graph
4.4
Long Division
set up the polynomials in a bracket, leading the factor outside
factor each variable
subtract and repeat
Remainder theorem
the remainder goes on top of the factor
Factor theorem
the polynomial (x-c) is a factor only if f(c)=0
Synthetic division
set up your polynomials in a row
multiply the first variables by the factor
subtract the answer by the next polynomial
repeat till you run out of polynomial
4.5
Fundamental theorem of algebra
for the polynomial of n is greater than of equal to 1 there is at least one complex zero
Real zero theorem
all polynomials of n have n amounts of complex zero
that is if their multiplicities are greater than one
Intermediate value theorem
if f(a)and f(b) have opposite signs then there is at least one row of zeros between a and b
Positive Descartes
the amount of positive zeros is equal to the number of variations in the signs
Negative Descartes
the amount negative zero is equal to the variations of signs
4.6
Rational Function
in form f(x)=g(x)/h(x),
Vertical asymptote
vertical line that the graph gets close to but never crosses
Horizontal asymptote
horizontal line the graph gets close to but never crosses
Properties horizontal asymptotes of a rational function
can have many vertical asymptotes but only one horizontal asymptoptes
can never intersect a vertical asymptote but can crosses a horizontal asymptote
Finding horizontal asymptotes of rational functions
if m is greater than n then there is a horizontal asymptote
if m is equal to n then the horizontal asymptote is y=a_n/b_m
if m is less then n there is no horizontal asymptotes
Removable discontinuities
a hole in the graph
when two functions share a common factor
Slant asymptote
when their is no vertical or horizontal asymptotes