To Graph a Function Completely
To Graph a Function Completely
|Characteristic |How to examine |
|A. 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: [pic], then y=a is an asymptote to the right |
| |[pic], then y=a is an asymptote to the left |
| |Vertical: [pic][pic], then x = a is an asymptote from the left |
| |[pic], 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
|f(x) FUNCTION |f ‘ (x) FIRST DERIVATIVE |f “ (x) SECOND DERIVATIVE |SKETCH |
|Increasing |Positive |No Clues here! | |
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|Decreasing |Negative |No Clues here! | |
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|Maximum |Will be = 0 or undef, AND Changing + to|Negative | |
| |- | | |
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|Minimum |Will be = 0 or undef, AND Changing – to|Positive | |
| |+ | | |
| | | | |
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|Concave Up |Increasing |Positive | |
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|Concave Down |Decreasing |Negative | |
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| | | | |
|Inflection Point |Maximum or Minimum |Will be = 0 or undef, AND Changing sign| |
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