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), then y = 1 / f(x) has a minimum point (a, 1/b)
If y = f(x) has a minimum point (a, b), 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
To sketch y = √f(x), note:
√f(x) is defined only if f(x) ≥ 0
√0 = 0 and √1 = 1
√f(x) < f(x) if f(x) > 1
√f(x) > f(x) if 0 < f(x) < 1
If y = f(x) has a stationary point (a, b), then y = √f(x) has a stationary point (a, √b)
if f(x) has a vertical asymptote at x = a, then so does y = √f(x)
If y = f(x) has a horizontal asymptote y = a, then y = √f(x) has a horizontal asymptote y = √a