**Summary**

To sketch y = | f(x) |, reflect the part of the graph of y = f(x) below the x-axis about the x-axis.

To sketch y = f( | x | ), ignore the part of the graph of y = f(x) on the left of the y-axis. Copy and reflect the part on the right of the y-axis about the y-axis.

To sketch y = 1 / f(x), note:

- if f(a) = 0, then y = 1 / f(x) has a vertical asymptote at x = a
- if f(x) has a vertical asymptote at x = a, then y = 1 / f(x) has an x-intercept at x = a
- 1/1 = 1 and 1 / (-1) = -1
- As f(x) increases, 1 / f(x) decreases
- As f(x) decreases, 1 / f(x) increases
- If y = f(x) has a maximum point (a, b) (where b ≠ 0), then y = 1 / f(x) has a minimum point (a, 1/b)
- If y = f(x) has a minimum point (a, b) (where b ≠ 0), then y = 1 / f(x) has a maximum point (a, 1/b)
- If y = f(x) has a horizontal asymptote y = a (where a ≠ 0), then y = 1 / f(x) has a horizontal asymptote y = 1/a
- If y = f(x) has an oblique asymptote, then y = 1 / f(x) has a horizontal asymptote y = 0

You can define functions using these operators: | ||||

+ | - | * | / | ^ |

abs( ) | sqrt( ) | ln( ) | exp( ) | pi |

sin( ) | cos( ) | tan( ) | ||

asin( ) | acos( ) | atan( ) |