Symmetry of a function, parity - odd and even functions
      Transformations of original or source function
         How some changes of a function's notation affect the graph of the function
         Translations of the graph of a function
         Reflections of the graph of a function
Symmetry of a function, parity - odd and even functions
A function ƒ that changes neither sign nor absolute value when the sign of the independent variable is changed is even, so that,  ƒ (x) = ƒ (-x).
Therefore, the graph of such a function is symmetrical with respect to the y-axis, as is the graph shown in the left figure below.
The graph of an even function. The graph of an odd function.
A function ƒ that changes sign but not absolute value when the sign of the independent variable is changed is odd, so that,        ƒ (x) = - ƒ (-x). That is, for each x in the domain of ƒƒ (-x) = -ƒ (x).  
Therefore, the graph of such a function is symmetrical with respect to the origin, as is the graph shown in the right figure above.
Transformations of original or source function
How some changes of a function notation affect the graph of the function
Some changes in a function expression (or an equation/formula) do not affect the shape or the form of the graph of the original or the given function y = ƒ (x).
Such changes include use of geometrical transformations to the graph of the function, like translations (or shifts) of the graph of the original function in the direction of the coordinate axes, or its reflection across the axes.
Translations of the graph of a function
The graph of a translated function  y = ƒ (x - x0) is obtained translating (shifting) the graph of its original or source function      y = ƒ (x) horizontally by x0 units to the right.
The graph of a translated function  y = ƒ (x) + y0  or  y - y0 = ƒ (x)  is obtained translating (shifting) the graph of its original function y = ƒ (x) vertically by y0 units up.
The graph of a translated function  y = ƒ (x - x0) + y0  or  y - y0 = ƒ (x - x0) is obtained translating (shifting) the graph of its original function y = ƒ (x) in both directions of the coordinate axes, horizontally by x0 units to the right and vertically by y0 units up.
Example:  Draw the graphs of the given three quadratic polynomials,
a)   y = x2 + 4x + 4  = (x + 2)2,       b)   y = x2 - 3     and    c)   y = x2 + 4x + 1  or   y + 3 = (x + 2)2
as the translations of the same source quadratic y = x2.
a)   y  = (x + 2)2 b)   y = x2 - 3 c)    y + 3 = (x + 2)2
Reflections of the graph of a function
Change of the sign of the independent variable of a function, denoted as y = ƒ (-x), reflects the graph of the given (original) function y = ƒ (x) across the y-axis.
Change of the sign of the function, denoted as  y = - ƒ (x), reflects the graph of the given function y = ƒ (x) across the x-axis.
Changes of the signs of both, independent variable and the function, denoted as y - ƒ (- x), reflect the graph of the given function  y = ƒ (x), across the y-axis and the x-axis.
Example:  Given quadratic polynomial  y = ƒ (x) = x2 + 4x + 1  or   y + 3 = (x + 2)2,  transform to:
                a)   y = ƒ (- x),     b)   y = - ƒ (x)     and     c)  y - ƒ (- x), and draw corresponding graphs.
Solution:  a)   y = ƒ (- x) = (-x)2 + 4(-x) + 1 = x2 - 4x + 1              or     y + 3 = (x - 2)2
                b)   y -ƒ (x) =  - (x2 + 4x + 1) = - x2 - 4x - 1             or     y - 3 = -(x + 2)2
                c)   y -ƒ (-x) =  - ((-x)2 + 4(-x) + 1) = - x2 + 4x - 1   or     y - 3 = - (x - 2)2
a)   y + 3 = (x - 2)2 b)   y - 3 = - (x + 2)2
c)   y - x2 + 4x - 1   or     y - 3 = - (x - 2)2
Calculus contents A
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