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Pre-Calc Exam 1 - Coggle Diagram

- - - - Logb(y) = x, y = b^x
      - No y intercepts, one x intercept
      - Inverse of an exponential function
      - Vertical asymptote at x = 0 in not transformed
      - Increasing at a decreasing Rate
      - No y intercepts, x intercept at (1, 0) if not transformed
  - - - A = P(1+r/n)^nt
      - Discrete Function
    - - A = Pe^rt
      - Continuous Function
    - - I = Prt
      - Discrete Function
    - - I = Interest
      - P = principle
      - A = Account Balance at the End of T years
      - R = Annualized Interest rate
      - T = Time in years
      - N = # of times compounded in a year
      - e = (1+1/n)^n as n approaches infinity, ~2.72
    - - Compound interest
        
        Present Value = A(1+r/n)^(-rt)
      - Continuous Interest
        
        Present Value = Ae^(-rt)
  - - - Exponential Growth
        
        b > 1
        
        Properties
        
        No x intercepts, y intercept at (0, a)
        
        Horizontal Intercept at y = 0 if it has not been transformed
        
        Increasing at an increasing rate
        
        The base is the consecutive ratio of outputs
      - Exponential Decay
        
        0 < b < 1
        
        Horizontal Intercept at y = 0 if it has not been transformed
        
        Decreasing at a decreasing rate
        
        No x intercepts, y intercept at (0, a)
        
        The base is the consecutive ratio of outputs
- - - - Domain: The set of all values that we can input to get a valid output.
      - Range: All outputs of a function
    - - Ex:) f(x) x g(x) = (f(x))(g(x))
      - Operations done on functions, and can be solved as polynomials
    - - Functions that are defined in parts, with one function defining one interval and another function defining another.
    - - Constant function: f(x) = b, where b is a real number
      - Identity function: f(x)=x
      - Square function: f(x)=x^2
      - Cube function: f(x)=x^3
      - Square root function: f(x)=sqrt.(x)
      - Cube root function: f(x)=cbrt.(x)
      - Reciprocal function: 1/x
      - Absolute value function: f(x)=|x|
  - - - Functions are odd when f(-x)=-f(x)
      - Functions are even when f(-x)=f(x)
    - - Local maxima are points where the graph of a function reaches its peak in an interval
      - Local minima are points where the graph of a function reaches its lowest in an interval
    - - The rate of change of a function in an interval is (f(b)-f(a))/(b-a)
- - - - Can be created by fitting a linear function to data (line of best fit)
    - - y=mx+b
        
        B is the value of f at f(0).
      - avg rate of change=slope (m)
        
        Increasing if m>0
        
        Decreasing if m<0
        
        Constant if m=0
  - - - Options for graphing a quadratic function:
        
        complete square to convert to vertex form
        
        sketch the function using transformations
        
        OR
        
        look at a to determine if the graph has a minimum or maximum
        
        Find the vertex and axis of symmetry
        
        find the y and x intercepts, if there are some
        
        find another point and then graph the function
    - - of the form ax^2+bx+c when a is not = to 0.
      - minimum vs maximum:
        
        minimum if a>0
        
        maximum if a<0
      - vertex form: a(x-h) ^2+k
      - vertex at (h,k)
        
        axis of symmetry: x=h line
      - discriminants
        
        if b^2-4ac>0, then the graph has two x intercepts
        
        If b^2-4ac=0, then the graph has one x intercept
        
        if b^2-4ac<0, then the graph has no x intercepts
- - - - A polynomial function can be thought of as the sum of monomials
      - it is of the form ax^n+ax^n-1+...ax+a0
      - A power function is a monomial of the form ax^n when a is not = 0 and n>0.
        
        Properties of power functions when n is even
        
        the graph is symmetric with respect to the y-axis, so it is even.
        
        Domain: all real numbers
        
        Range: all non-negative real numbers
        
        Graph always contains points (0,0), (1,1) and (-1,1)
        
        properties of power functions when n is odd
        
        the graph is symmetric with respect to the origin, so it is odd
        
        the domain and range are both all real numbers
        
        Always contains the points (0,0), (1,1), and (-1,-1)
      - even vs odd multiplicity
        
        even: graph touches x axis
        
        odd: graph crosses x axis
      - The maximum amount of turning points in a polynomial with degree n is n-1.
    - - X-intercepts
      - Found most easily when fully factored
      - if (x - r) is a factor, x = r is a root
  - - - A function of the form p(x)/q(x)
      - vertical asymptotes found by finding real zeros of the denominator
      - a function is proper if the degree of the numerator is less than the degree of the denominator. if this is the case, then y=o is a horizontal asymptote.
      - other horizontal or oblique asymptotes are found by polynomial long division.