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Advanced Functions Midterm
Unit One - Functions Review
Lesson One - Factoring
There are 4 types of factoring
Simple Trinomial (adds to multiplies to)
Difference of squares ( 9x^2-4)
Complex Trinomial (criss cross, Australian and, Decomposition)
Greatest common factoring
Lesson Two - Familiar Functions
A relation is: a relationship between two or more variables
A function is: a relation where each x value only has one y value
A relation is a function if it passes the vertical line test
Lesson Three - Transformations |
y= af(b(x-c)) + d
A: if -ve reflect of the x axis, vertical stretch by a factor of a (a > 1), vertical compression by 1/a (0<a<1)
B: Horizontal stretch by a factor of 1/b (0<b<1), horizontal compression by b (b>1) if -ve reflect of y-axis
C: horizontal shift right “c” units (-), horizontal shift left “c” units (+)
D: Vertical shift up “d” units (+), vertical shift down “d” units (-)
Lesson Five - Inverse Functions
To find the inverse of a function algebraically:
Switch the x and y in the equation
Solve for the new y
Rename y to be the inverse of the original function given
Find the inverse of a function graphically
switch the x and y coordinates of each point on the graph. the inverse will be a reflection of the original function off the line y=x
Invariant points are the points that did not change
The inverse of a function is not necessarily a function
Unit Two - Polynomials |
Lesson One - Intro to Polynomial Functions
A polynomial function is a function that can be expressed in the form: f(x)=ax^n+bx^n−1+cx^n−2+…+dx^3+ex^2+fx+g
N is the degree of the function, and must be a whole number
X is a variable
A, B, C, D are all coefficients
A is the leading coefficient
Specific degrees
0 = constant
1 = linear
2 = quadratic
3 = cubic
4 = quartic
5= quintic
Symmetry
Line (even) - You can divide the graph into two equal parts along the y-axis
Point (odd) - The graph can rotate 180 and the graph will stay the same
Even and Odd degree Functions
Even and positive - Start high, end high
Even and negative - start low, end low
Odd and positive - start low, end high
Odd and negative - start high, end low
Lesson Three - Factored form of Polynomial Functions
Orders:
1 - straight through
2 - bounce
3 - squiggle
Symmetry
Even degree functions are not necessarily even functions
Odd degree functions are not necessarily odd functions
An even function is symmetric about the y-axis. f(-x) = f(x)
A function that is odd is symmetric about the origin. f(-x) = -f(x)
Lesson Five - Slope of Secant vs. Tangent
The secant is a line that crosses a curve at two points
the tangent is a line that touches a curve at one point
Slope of a secant = average rate of change (ARC)
sloe of a tangent = instantaneous rate of change (IRC)
Lesson Six - Instantaneous rates of change
Three ways to determine the IRC
Graphing: graph the function, draw the tangent estimate 2 points on the line, find the slope.
Substituting: use the given x1 and y1 then sub in a value that is extremely close to the two values, use the slop formula, do not round
Algebraically: use the first principles formula
Unit Three - Polynomials ||
Lesson One - Remainder Theorem
If the remainder is 0, this means that the divisor is a factor of the dividend
if the remainder is not zero, the divisor is not a factor of the dividend
Lesson Four - Family of Polynomial Functions
Family of functions = a set of functions that have the same characteristics (same x-intercepts but different y-intercepts
when making an equation add on, KER, K can’t = 0
Lesson Five - Solving Inequalities Graphically
On a number line:
Closed dot = right on the number
Open dot = doesn’t hit the number
Unit Four - Rationals
Lesson One - Introduction to Rational Functions
The reciprocal of a function is: 1/the function
Invariant points: where f(x)= +/-1
Local max/min: Wherever f(x) has one, g(x) will too)
Lesson Two - Limits
We say that the limit of a function f(x), as x approaches a point a, is equal to a number (call is L). If f(x) can be made as close to L as desired by making x sufficiently close to a, but not equal to a. If there is no such number we say that the limit does not exist
Lesson Three - Vertical and Horizontal Asymptotes
Vertical Asymptotes:
Occur where a function is undefined (denominator is equal to zero)
To find them, factor the denominator then set each factor to zero and solve
Horizontal Asymptotes
Can be cross the HA somewhere in the graph
degree in numerator = degree in the denominator, LCn/LCd
degree in numerator < degree in the denominator HA = 0
degree in numerator > degree in the denominator HA = oblique (found through long division)
Lesson Six - Instantaneous and Average Rates of Change
The formulas can be used for rational functions; however, the interval must not include where the function is undefined
Advanced Functions Midterm
Unit One - Functions Review
Lesson One - Factoring
There are 4 types of factoring
Simple Trinomial (adds to multiplies to)
Difference of squares ( 9x^2-4)
Complex Trinomial (criss cross, Australian and, Decomposition)
Greatest common factoring
Lesson Two - Familiar Functions
A relation is: a relationship between two or more variables
A function is: a relation where each x value only has one y value
A relation is a function if it passes the vertical line test
Lesson Three - Transformations |
y= af(b(x-c)) + d
A: if -ve reflect of the x axis, vertical stretch by a factor of a (a > 1), vertical compression by 1/a (0<a<1)
B: Horizontal stretch by a factor of 1/b (0<b<1), horizontal compression by b (b>1) if -ve reflect of y-axis
C: horizontal shift right “c” units (-), horizontal shift left “c” units (+)
D: Vertical shift up “d” units (+), vertical shift down “d” units (-)
Lesson Five - Inverse Functions
To find the inverse of a function algebraically:
Switch the x and y in the equation
Solve for the new y
Rename y to be the inverse of the original function given
Find the inverse of a function graphically
switch the x and y coordinates of each point on the graph. the inverse will be a reflection of the original function off the line y=x
Invariant points are the points that did not change
The inverse of a function is not necessarily a function
Unit Two - Polynomials |
Lesson One - Intro to Polynomial Functions
A polynomial function is a function that can be expressed in the form: f(x)=ax^n+bx^n−1+cx^n−2+…+dx^3+ex^2+fx+g
N is the degree of the function, and must be a whole number
X is a variable
A, B, C, D are all coefficients
A is the leading coefficient
Specific degrees
0 = constant
1 = linear
2 = quadratic
3 = cubic
4 = quartic
5= quintic
Symmetry
Line (even) - You can divide the graph into two equal parts along the y-axis
Point (odd) - The graph can rotate 180 and the graph will stay the same
Even and Odd degree Functions
Even and positive - Start high, end high
Even and negative - start low, end low
Odd and positive - start low, end high
Odd and negative - start high, end low
Lesson Three - Factored form of Polynomial Functions
Orders:
1 - straight through
2 - bounce
3 - squiggle
Symmetry
Even degree functions are not necessarily even functions
Odd degree functions are not necessarily odd functions
An even function is symmetric about the y-axis. f(-x) = f(x)
A function that is odd is symmetric about the origin. f(-x) = -f(x)
Lesson Five - Slope of Secant vs. Tangent
The secant is a line that crosses a curve at two points
the tangent is a line that touches a curve at one point
Slope of a secant = average rate of change (ARC)
sloe of a tangent = instantaneous rate of change (IRC)
Lesson Six - Instantaneous rates of change
Three ways to determine the IRC
Graphing: graph the function, draw the tangent estimate 2 points on the line, find the slope.
Substituting: use the given x1 and y1 then sub in a value that is extremely close to the two values, use the slop formula, do not round
Algebraically: use the first principles formula
Unit Three - Polynomials ||
Lesson One - Remainder Theorem
If the remainder is 0, this means that the divisor is a factor of the dividend
if the remainder is not zero, the divisor is not a factor of the dividend
Lesson Four - Family of Polynomial Functions
Family of functions = a set of functions that have the same characteristics (same x-intercepts but different y-intercepts
when making an equation add on, KER, K can’t = 0
Lesson Five - Solving Inequalities Graphically
On a number line:
Closed dot = right on the number
Open dot = doesn’t hit the number
Unit Four - Rationals
Lesson One - Introduction to Rational Functions
The reciprocal of a function is: 1/the function
Invariant points: where f(x)= +/-1
Local max/min: Wherever f(x) has one, g(x) will too)
Lesson Two - Limits
We say that the limit of a function f(x), as x approaches a point a, is equal to a number (call is L). If f(x) can be made as close to L as desired by making x sufficiently close to a, but not equal to a. If there is no such number we say that the limit does not exist
Lesson Three - Vertical and Horizontal Asymptotes
Vertical Asymptotes:
Occur where a function is undefined (denominator is equal to zero)
To find them, factor the denominator then set each factor to zero and solve
Horizontal Asymptotes
Can be cross the HA somewhere in the graph
degree in numerator = degree in the denominator, LCn/LCd
degree in numerator < degree in the denominator HA = 0
degree in numerator > degree in the denominator HA = oblique (found through long division)
Lesson Six - Instantaneous and Average Rates of Change
The formulas can be used for rational functions; however, the interval must not include where the function is undefined