To Graph a Function Completely
Characteristic / How to examineA. Domain / Domain is usually restricted by denominators that equal 0 and by radicals that contain negative values. Sometimes there may be an explicit domain.
B. Intercepts / y-intercepts are found by computing f(0).
x-intercepts are found by solving f(x) = 0
C. Symmetry / Even: f(-x)=f(x) will have y-axis symmetry
Odd: f(-x)=-f(x) will have origin symmetry
Periodic: trig functions will repeat over a certain period of x-values
D. Asymptotes / Horizontal: , then y=a is an asymptote to the right
, then y=a is an asymptote to the left
Vertical: , then x = a is an asymptote from the left
, then x = a is an asymptote from the right
Slant: Perform long division, and ignore the remainder
E. Increasing/Decreasing / f(x) is increasing whenever f ‘ (x) is positive
f(x) is decreasing whenever f ‘ (x) is negative
F. Extrema / f(a) is a maximum if f ‘ (a) is 0 or undefined, AND f(x) is changing from increasing to decreasing (this would show up as a sign change on f ‘(x) from positive to negative, and also as a negative value for f “ (a) Also, any closed endpoints must also be examined for potential maximums.
f(a) is a minimum if f ‘ (a) is 0 or undefined, AND f(x) is changing from decreasing to increasing (this would show up as a sign change on f ‘(x) from negative to positive, and also as a positive value for f “ (a) Also, any closed endpoints must also be examined for potential minimums.
G. Concavity / f(x) is concave up whenever f “ (x) is positive. (this would also show up as f ‘ (x) increasing.)
f(x) is concave down whenever f “ (x) is negative. (this would also show up as f ‘ (x) decreasing.)
H. Inflection Points / (a, f(a) ) is an inflection point if f(x) is changing concavity at a. This would show up as a sign change on f “ (x), in either direction.
I. SKETCH CAREFULLY / If needed, calculate a few y-values by substitution
* Verify the sketch / Check the points and/or intervals for each characteristic to make sure your sketch matches your analysis, Be sure you have labeled the scale on each axis, and that your graph and work are legible.
How f(x) , f ‘ (x), and f “ (x) are related
Increasing / Positive / No Clues here! /
Decreasing / Negative / No Clues here! /
Maximum / Will be = 0 or undef, AND Changing + to - / Negative /
Minimum / Will be = 0 or undef, AND Changing – to + / Positive /
Concave Up / Increasing / Positive /
Concave Down / Decreasing / Negative /
Inflection Point / Maximum or Minimum / Will be = 0 or undef, AND Changing sign /