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AP Calculus AB Unit 1 Project Mind Map (Covers the Entirety of 1.1-1.6) …

- - - - These are functions that are combined together. The output of one function can be used as the input of the other function(s).
      - Example of a composite function expression: h = g ∘ f
        
        In this case, it is basically h(x) = g(f(x)). The input of function g is the output of function f, which then creates function h
      - Related
    - - Equations
        
        Absolute Value Equation:
        f(x) = a|x - h| + k
        
        Example:
        y = |x|
        For this, the line bends at points (0, 0) upward with a slope of 1. The y cannot be negative, so the range is y ≥ 0, and domain is just xER.
    - - Even
        
        End behavior extends from quadrant 2 to quadrant 1 or quadrant 3 to quadrant 4
        
        Equation:
        f(-x) = f(x)
      - Odd
        
        End behavior extends from quadrant 3 to quadrant 1 or quadrant 2 to quadrant 4
        
        Equation:
        f(-x) = -f(x)
      - Related
    - - Logarithmic Functions
        
        Logarithmic function is the inverse of exponential function
      - Exponential growth:
      - Exponential growth is growth which multiplies and its growth increases as time goes on. There will be asympote
      - When a quantity decays (opposite of growth) by a multiple and its eventually it will reach a limit during which the decay will be virtually nothing, but there will be asymptote.
      - Exponential decay:
      - Example equation: y = 2^x
    - - Graph is reflected over the y-axis
        
        In an equation, just switch the places of x and y and solve from there
      - Related
      - Related
    - - Horizontal Line Test
        
        If it passes the horizontal line test, then it is a one to one function
        
        If it doesn't pass, then it is not a one to one function
        
        Basically like the vertical line test, except that the line can only cross the horizontal line once.
      - The domain has its distinct elements. The codomain of the function only will have one domain assigned to it
      - Using this method to see if the function is one to one:
    - - Unit Circle
      - Sine Function
        
        Sine functions have a period of 2π
        
        Equation
        y = Asin(A(x - C)) + D
        
        The first A changes the amplitude
        
        The second A changes the period
        
        The C is responsible for phase shifting (left or right)
        
        The D is responsible for vertical shifting (up or down)
      - Cosine Function
        
        Cosine functions have a period of 2π
        
        Equation
        y = Acos(A(x - C)) + D
        
        The first A changes the amplitude
        
        The second A changes the period
        
        The C is responsible for phase shifting (left or right)
        
        The D is responsible for vertical shifting (up or down)
      - Tangent Function
        
        Tangent functions have a period of π
        
        Equation
        y = Acos(A(x - C)) + D
        
        The first A changes the amplitude
        
        The second A changes the period
        
        The C is responsible for phase shifting (left or right)
        
        The D is responsible for vertical shifting (up or down)
- - - - Linear regression
        
        Using line algorithm for AI (ŷ = a + bx)
        
        Using input and output method to obtain results
        
        Features and labels
        
        In the real world, it can be used for sales, prices, GDP, etc. (mostly constant rates)
      - Logistic regression
        
        The sigmoid function (most popular)
        
        This can be used to predict if students are qualified to get in a university, the output being accepted or rejected, and the inputs being multiple factors, like GPA, extra curricular activities, etc.
        
        Why do we use sigmoid function?
        It works only when it is a binary classification problem
        They have to be chosen so it makes the best fit regression, which makes the smallest mean squared error
        Used for binary classification only because, as shown in the diagram, the y value can either be 0 or 1, which is binary. The 0 or 1 can represent yes or no, positive or negative, pass or fail, etc, respectively.
        
        Some data sets have a general behavior that fits the sigmoid function, which is used in logistic regression
        
        Tanh function
        
        This is usually used in neural networks, within the hidden layers. Simply speaking, it helps the data center by bringing the mean closer to a center point
        
        The difference between tanh and tan is tanh is a hyperbolic tangent function. It is equivalent to sinh x / cosh x, and is an odd function. Also, the tanh function is a one to one function
        
        ReLU function
  - - - The form is: ŷ = a + bx
        
        b is the slope, x is the x coordinate and a is the y-intercept. The y hat is the predicted value of y.
        
        This relationship is basically displaying the relationship between data of two variables
      - It is useful in the real world to observe or predict values using the line of best fit, by inputting an x value (or y) to get the other using the predictions made by the line.
        
        For example, a line of best fit is created to predict the y-value (height) of an infant using time after birth (in weeks, x). A equation will be formed: ŷ = a + bx, with given values, and using x (time) to predict y (height of infant)
    - - Equation:
      - The normal curve has points of inflection exactly one standard deviation from the center, and the area under the curve has to equal to one, which represents 100%. This function is bell shaped.
        
        Diagram:
        
        This is very useful in the real world, as many populations behave normally, with a mean and standard deviation. Standardized assessments, like the SAT, also generally has a normal distribution of scores from 0-1600
  - - - Example:
        
        This is using the exponential function to graph and visualize the relationship of a Carbon-14 decay
      - These predictions are useful in lab scenarios. It keeps the lab experimenters safe, as they can see from a exponential decay graph when the decay can go into a dangerous state, especially with alpha particles
    - - As shown from all these real-world applications, using graphs and functions are extremely pragmatic in the developing world.