Symmetry and Coordinate Graphs

A function is symmetric around a line if the graph on one side of the line is the mirror image of the graph on the other side of the line.

Reflections across the x-axis:

• y = - f(x)        Take f(x) and draw its mirror image across the x-axis (turns the graph upside down).
  
• y = |f(x)|         Take the parts of f(x) that are under the x-axis and draw their mirror
                         images above the x-axis.  Leave the parts of f(x) that are above the x-axis
                         where they are.

Reflections across the y-axis:

• y = f(-x)          Take f(x) and draw its mirror image across the y-axis (rotates the graph leftto right and
                         right to left).
• This is called an EVEN function.
• To test if a function is even, show that f(-x) = f(x).

Reflection across the line y = x:

• x = f(y)            Take f(x) and draw its mirror image across the line y = x (the two functions
                          are inverses of each other).

Symmetry around the origin:

• A function is symmetric around a point if a line can drawn through the point and extended until it reaches the function on both sides so that the line is bisected by the point.
• This is called an ODD function
• To test if a function is even, show that f(-x) = -f(x)

Axis of Symmetry:

• For a quadratic:  x = -b/2a
• For a cubic:  x = -b/3a

Basic Toolkit of Functions and Relations:

You should know these by name, equation, and graph.  You should also know their domains and ranges.

1. Constant:  f(x) = a     
      Domain:  Reals
      Range:  a
2. Identity:  f(x) = x
      Domain:  Reals
      Range:  Reals
3. Reciprocal:  f(x) = 1/x
      Domain:  x is not 0
      Range:  y is not 0
4. Quadratic:  f(x) = x²
      Domain:  Reals
      Range:  y > 0
5. Power:  f(x) = xⁿ
      n is even                    n is odd
      Domain:  Reals          Domain:  Reals
      Range:  y > 0             Range:  Reals
6. Cubic:  f(x) = x³
      Domain:  Reals
      Range:  Reals
7. Greatest Integer:  f(x) = [x]  (Floor Function)
      Domain:  Reals
      Range:  Integers
8. Square Root:  f(x) = \/x
      Domain:  Reals
      Range:  y > 0
9. Absolute Value:  f(x) = |x|
      Domain:  Reals
      Range:  y > 0
10. Exponential:  f(x) = a^x
       Domain:  Reals
       Range:  y > 0
11. Logarithmic:  f(x) = log_ax
       Domain:  x > 0
       Range:  Reals
12. Trigonometric:  f(x) = sin x               f(x) = cos x                 f(x) = tan x
                           Domain:  Reals          Domain:  Reals            Domain:  x is not (2a+1)(pi)/2
                           Range:  -1 < x < 1     Range:  -1 < x < 1       Range:  Reals
13. Semicircle:  f(x) = \/(a² - x²)
       Domain:  -a < x < a
       Range:  0 < x < a
14. Polynomial:  f(x) = (x - a)(x - b)(x - c)...
       Domain:  Reals
       Range:  Varies
15. Circle:  x² + y² = r²
       Domain:  -r < x < r
       Range:  -r < x < r
16. Hyperbola:  x² - y² = 1
                      a²    b²
       Domain:  x > a, x < -a
       Range:  Reals
17. Ellipse:  x² +  y² = 1
                a²       b²
       Domain:  -a < x < a
       Range:  -b < x < b
    

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