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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