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)