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AP Calculus AB Unit 1 Project Mind Map (Covers 1.1-1.6) By: Kevin Virk…

- - - - If the function is plotted as a graph then you can check the following to see whether it is even or odd
        Even Functions: Reflected on the y axis
        Odd Functions: Symmetrical on Origin
      - End Behaviour
        Even functions will go from Quadrant I -> II or Quadrant III -> IV
        Odd Functions will go from Quadrant I -> III or Quadrant II -> IV
    - - To completely verify whether something is a even or odd function algebra can be used. Graphically verifying is good but algebraically is gives you a definite answer
        Even Functions: f(-x)=f(x)
        Odd Functions: f(-x)=-f(x)
  - - - Function Machine
      - Table of Values
        A chart which helps to show a set of ordered pairs and can be later turned into a graph
      - Equations
      - Mapping
      - Graphing
    - - Graphically: Vertical Line Test
        If there is a curve and you place a vertical line anywhere in the graph and that vertical line does not intersect more than one time, the graph is a function
        
        In this example, the circle on the left is a relation while on the curve on the right is a function
      - What would this mean: Relation vs Function
        A relation would be considered as any set of ordered pairs due to which any function can be a relation. However not any relation is a function as a function is a relation in which there is a unique y present for every value of x
        
        You can see this by mapping the relations and seeing if the ordered pairs have unique x values. However the y can have many x values
        
        X cant have more than one y value or else it fails the Vertical Line Test
- - - - Slope
        Is associated with the coefficient "m" and is used to know a line's incline. The formula for a slope is ʸ²⁻ʸ¹⁄ₓ₂₋ₓ₁ or as it is commonly referred to as, "rise over run"
        
        Perpendicular Lines
        Two lines can be considered as perpendicular if the slopes of both lines are the negative reciprocal of each other (m1 x m2 = -1)
        
        In this example there are the lines y=3x+2 and y=-1/3x+10/3 which are perpendicular to each other
        
        Fun Fact
        There is one solution to the system of equations that the lines represent as there is only one intersection
        
        Slope of 0
        
        Horizontal Line
        
        Undefined Slope
        
        Vertical Line
        
        Parallel Lines
        Two lines can be considered as parallel if the slopes of both lines are identical
        (m1=m2)
        
        In this example there are the lines y=3x+8 and y=3x+4 which are parallel to each other
        
        Fun Fact
        There are no solutions for the system of equations that the lines represent as they will never intersect
        
        Coincident Lines
        When there are lines that have the same slope and y-intercept thus are on top of each other but appear to look like one line
        (m1=m2) (b1=b2)
        
        In this example both lines are y=9/2x+8
        
        Fun Fact
        There are infinite solutions for the system of equations the lines represent as they with "intersect" infinitely
- - - - How to identify
        Graphically: If there seems to be an increase or decrease of a quantity as the x value changes. This is simple to spot
        The B Value, as previously stated can also be seen to see whether a there is growth or decay
        A Table of Values can also be utilized to recognize if there has been growth or decay as the numbers will increase/decrease
- - - - The coefficient in the values in the ordered pair is 2
  - - - Parametrizing a line segment
        If you have two points which make a line segment, you can parametrize that to get the parametric equations. To do this, you would find your two points (x1,y1) and (x2,y2) and then assign one as T=0 and another as T=1. Then using the Formula of x(t)=aT+B and y(t)=cT+D, you can substitute T=0 to get the D value. Afterwards you can substitute for T=1 in those equations and get x2=aT+B and y2=cT+D and solve for a and c to get your final equations in the Formula that was first stated. You do not have to use T=0 and T=1 however that is the most simple method.
        
        In this example, points of (2,1) and (3,5) were parameterized and using the method explained we get x(t)=t+2 and y(t)=4t+1
- - - - Function A passes the test and is one-to-one
        Function B does not and is just a function