Extrema of Polynomials

Extrema

Extrema, or critical points, are maximum or minimum values of a function.

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--> They can usually be spotted by looking for places where the function changes direction, or bends

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--> Using a graph is the BEST strategy for finding extrema​

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The highest/lowest point on a specific interval or in a specific graph window.

These should always have numerical or coordinate values.​

​Local/Relative

The highest/ lowest point of the whole function (including off the graph).​

If a function's end goes to infinity, these count as an absolute extrema​

​Global/Absolute

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Identifying Function Types

You can use the number of extrema to help identify your function type, and vice versa

-->Take the number of extrema and add one. This is the LOWEST possible degree of your function.

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--> Take the degree (highest exponent) and subtract one. This is the HIGHEST number of extrema you could have.​

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Polynomials SO FAR...

• End Behavior

• Zeroes

• Extrema​

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Special Extrema

• Imaginary: the function changes direction WITHOUT CROSSING the x-axis
  

• Multiplicity: the function changes direction EXACTLY on the x-axis (represents 2, 4, or 6 zeroes in the same place)

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The number next to the highest degree term.​

POSITIVE: points up or trends upward​

NEGATIVE:​points down or trends downward

​Coefficients

EVEN degrees have ends that point in the same direction.

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ODD degrees have ends that point away from each other​

​Degree

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