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Math (gr.12 advanced Functions)
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Terms in this set (71)
How can I find Range
Look left to right
How to find Domain?
Look up and down
Interval of increase
Intervals within a functions domain where the y values of the function get larger (moving left to right)
Interval of decrease
Intervals within a function domain where the y values of the function get smaller moving from left to right
Continuous functions
Any function that does not contain any holes or breaks over its entire domain
Discontinuous function
Any function that contains a break in its graph
Interval where function Is positive
Interval where all y-vallues are above the x axis
Interval where function Is negative
Interval where all y values are below the x axis
Transformation formula
Y=af(k (x-d))+c
What happens when a in transformation is negative
Reflection in y axis
What happens when k is negative in transformation
Reflection in x axis
Difference between polynomial function and polynomial expression
Polynomial expression does not have y=
Characteristics of polynomial graphs
Domain is not restricted . Range depends. Can have either upper or lower bounds NOT BOTH
Polynomial with a negative odd degree (end behaviors)
Left side -up right side- down
Polynomial with positive odd degree (end behaviors )
Lefts side - down right side - up
Polynomial with negative even degree (end behaviors )
Both down
Polynomial with positive even degree (end behaviors)
Both up
Polynomial with a positive coefficient (end behaviors )
Both up
Polynomial with negative coefficient (end behaviour )
Both down
Polynomial turning points formula
N-1 (n being the degree)
Polynomial with even degree has how many zeros
From 0 Up to n amount (n being degree )
Polynomial with odd degree has how many zeros
At least 1 to n amount (n being degree)
How to graph polynomial in factored form
1) plot zeros
2) determine degree & if even or odd
3) determine end behaviors
If polynomial has a squared factor what happens to graph
The factor forms quadratic shape on x-axis (U shape)
If polynomial has a cubed factor what happens to graph
The factor forms cubic shape through x-axis (S shape)
Characteristics of an even function
Symmetric about the y axis . All exponents are even. When put -x into x, the signs stay the same.
Characteristics of an odd function
Symmetric about the origin . All exponents are odd. When put -x into function signs change.
Characteristics of a "neither " graph (not even or odd)
No symmetry. Exponents are mixed. When put -x into graph signs are still mixed.
What is the order of a division statement
Dividend.= divider x quotient +remainder
How can you tell if a divisor is a factor ?
If once put into the equation, there is no remainder
How to factor polynomials of degree 3 or higher
1) use factor theorem to find a number that when substituted into f (x) will equal zero
2) use synthetic division
3) keep factoring till done
How to SOLVE polynomial functions
To solve make polynomial equal zero, then solve normally
How can u solve a polynomial inequality
1) factor to find zeros
2) sketch using zeros and end behaviors
3) determine intervals that satisfy inequality
4) provide solution on NUMBER LINE
When do you use open and closed dot on a number line
Closed dot : _< or >_
Open dot : < or >
What's the rules for solving polynomial inequalities graphically
When dividing by a negative the equality sign changes. Always put in interval notation
What is a hole and how to find them
Break in graph . Occurs when f(x) and g(x) have a common factor (so they cancel out)
How to find a horizontal asymptote
CAN BE CROSSED. Degree of numerator is one less than the degree of the denominator.
If the degrees are the same, divide the leading coefficient this becomes HA.
How to find a vertical asymptote
Make denominator equal zero and solve
How to find an oblique asymptote
Degree in numerator is greater than degree in denominator by EXACTLY ONE.
Now to find x intercept in rational function
Allowing numerator to equal zero.
How to find y intercept in rational function
Let all x's equal zero and solve
How to graph rational functions
Factor
Determine intercepts
Determine asumptots
IF HA HAVE TO TEST END BEHAVIORS
Steps for solving rational functions
State restrictions on x
Add, subtract until there is only one expression on each side of the equal sign
Cross multiply
Solve and check if any solutions should be rejected
How to solve rational inequalities algebraiclly
Move everything to the left side of the inequality sign and simplify
Determine VA AND zeros
Use table with test values to determine solution
What is a radian
A measure of an angle using the unit circle
How to convert from degrees to radian
Multiply angle by PIE/180
To go from radians to degrees?
Multiply 180/PIE
What's the difference between exact value and approximate value
Exact value uses the pie and approximate value is the decimal
What are the special angles on the unit circle?
PIE/3 = 1.047
PIE/4 = 0.785
PIE/6= 0.524
Four quadrants on unit circle in radians
90° (PIE/2)
180° (PIE)
270°(3PIE/2)
360° (2PIE)
Special angle triangles PIE/3
Adj: 1
Opp: root 3
Hyp: 2
Special angle triangle PIE / 4
Adj: 1
Opp: 1
Hyp: root 2
Special angle triangle PIE/6
Adj: root 3
Opp: 1
Hyp: 2
Define amplitude
Vertical stretch or compression (A)
Define period
Horizontal stretch or compression ( K)
Define phase shift
Horizontal shift left or right (D)
Define equation of shift
Vertical shift up or down (C)
Formula for finding period
2PIE/K
Formula for amplitude
Max-min/2
What are the cos function identities
Sinx= cos (PIE/2-x)
Cosx = sin (PIE/2-X)
tan = cot (PIE/2- X)
Properties of log function
D is greater NOT EQUAL TO zero
R not restricted
VA at zero
X int at (1,0)
How to graph basic log functions
Look at number of base
Two points are (base, 1) and (1/base,-1)
When are log functions increasing or decreasing
A greater than zero = increasing
A is a fraction = decreasing
General log equation ?
Alogk (x-d)+c
How to find x int of transformed log function
Use x int of general log function (1,0)
How to find VA of log function using equation?
It's the D, make it equal zero and solve
How to find Domain of log function
It's the VA
Make d equal zero and solve
General log equation ?
Alogk (x-d)+c
How to find x int of transformed log function
Use x int of general log function (1,0)
How to find VA of log function using equation?
It's the D, make it equal zero and solve
How to find Domain of log function
It's the VA
Make d equal zero and solve
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