MM

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

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:

  1. Switch the x and y in the equation
  2. Solve for the new y
  3. Rename y to be the inverse of the original function given

Find the inverse of a function graphically

  1. 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
  2. Invariant points are the points that did not change
  3. 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

  1. Graphing: graph the function, draw the tangent estimate 2 points on the line, find the slope.
  2. 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
  3. 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