Functions

Functions

1. Types of Parent functions

   • Cosine
     • infinite number of zeros
   • Sine
   • Exponential
   • Absolute Value
     • deals with magnitude of coordinate, ouput of absolute function always = (+) value
     • absolute notation: intervals where all real numbers are expressed by notation and represented on number line ex. |x| > 2
   • Square Root
   • Linear
   • Reciprocal
     • \[2^x\] 
     • asymptote = a line or curve that comes close but doesn't touch the certain line ex.
       • x = 0 y = 0
       • asymptote: y = 0
   • Quadratic

2. Transformations

   • Y = a f[k(x-d)] + c
     • a value
       • a > 1 = vertical stretch by a factor of 'a'
       • 0 < a < 1 = vertical compression by a factor of 'a'
       • -a = relfection on x-axis
     • k value
       • k > 1 = horizontal compression by a factor of ' 1 / k'
       • 0 < k < 1 = horizontal stretch by a factor of ' 1 / k'
       • -k = reflection on y-axis
       • affects x-value (inside bracket = opposite)
         • use to find new (x & y) points for transformed function graph
     • c value
       • c < 0 = vertical translation down
       • c > 0 = vertical translation up
       • affects y-value
       • c value = y intercepet (initial value)
     • d value
       • d < 0 = horizontal translation 'd' to the left
       • d > 0 = horizontal translation 'd' to the right
       • gives x-intercept

3. Inverse

   • f(x) = f-1(x)
     • steps: 1. switch y and x variables 2. isolate y and get inverted function
   • reflected y=x shows original and inverted functions

4. Terms

   • Properties
     • end behaviour: at ends of function where y values approach x
     • odd
       • f(-x) = -f(x) *opposite function
       • symmetrical when rotated 180 about origin
     • turning point; point on function where it changes from decreasing to increasing & vice. versa
     • interval of decrease: y-values get smaller moving left to right on domain interval of inecrease: y-values get larger moving left to right on the domain
     • even
       • symmetrical about y-axis
       • f(-x) = f(x) *function stays the same
     • Domain: affects x-axis
     • Range: affects y-axis
   • Characteristics
     • Vertical line test
       • every x-value has ONLY 1 y-value
     • continuous/ discontinuous
       • any breaks or gaps in function?
       • piecewise function
         • function notation: algebraically expresses piecewise *open dot=not including closed dot=include ex.
         • function that is made up of pieces of different or similar function

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