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Chapter two - Coggle Diagram

- - - - domain=all the real numbers excluding the real zeros of 𝑞(𝑥).
  - - - f(x) approaches infinity or negative infinity as x approaches c
  - - - to find= divide the numerator of the function by the denominator
      - IMPORTANT: 1) The graph of a rational function may crossits slant asymptote. 2) The graph of a rational function may have a horizontalasymptote ora slantasymptote but not both.
- - - - f(x)=anx^n+an-1x^n-1+an-2x^n-2+...+a1+a0
      - p/q=Factors of a0c(constant terms)/ Factors of an (leading coefficients)
        
        = rational zero exists for a polynomial
  - - - f(x) hasat least one complex zero
  - - - c1, c2, ... cn are complex numbers.
  - - - a–biis also a zero of f(x
  - - - f(x) has exactly no complex zeros
- - - - no sharp points
    - - Sharp Point
    - - Break in graph
    - - Hole
      - Break in graph
  - - - x approaches infinity
        
        x becomes infinitely large in the positive direction
    - - x approaches negative infinity
        
        x becomes infinitely in the negative direction
    - - f(x) approaches infinity
        
        the y value infinitely increases in the positive direction
    - - f(x) approaches negative infinity
        
        the y value infinitely increases in the negative direction
  - - - both arrows go upwards
    - - both arrows go downward
  - - - left arrow downward right arrow upward
    - - left arrow upward and right arrow downward
- - - - f(x) is the dividend
        
        q(x) is the quotient
    - - d(x) is the divisor
        
        r(x) is the remainder
  - - - remainder = f (c)
    - - f(c) = 0
        
        (x–c)= factor of f(x).
- - - - Linear
        
        Degree:1
        
        General form: ax+b
      - Quadratic
        
        Degree:2
        
        General form: ax^2+bx+c=0
        , a does not equal 0
      - Cubic
        
        Degree:3
        
        General form: ax^3+bx^2+c=0
        ,a does not equal 0
- - - - f(X)=ax^2+bx+c, a does not equal 0
        
        Vertex Form f(x)=a(x-h)^2+k
        
        Vertex: (h,k)
      - parabola
      - opens upward for a> 0
      - opens downward for a< 0
- - - - f(c) is positive
        
        values of x on the interval are solutions to f(x) > 0
      - f(c) negative
        
        values of x on the interval are solutions to f(x) < 0