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Properties of FunctionsSection 3.3

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Even Functions: Notice here that .

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Odd Functions: Notice that .

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Even, Odd or Neither?

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Identify Even/Odd/Neither From Equation1. Exercise 35 Page 238 2. Exercise 37 Page 238 3. Exercise 42 Page 238

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Recall Interval NotationThe following interval contains 3 parts:

• Values less than or equal to -6 : • Values greater than -2 but less than or equal to 2: • Values greater than 4: Put them all together :

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Increasing and Decreasing

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Where is the graph increasing/decreasing?

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Where is the function inc/dec?

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Where is the function inc/dec?

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Local Maxima and MinimaThese are the values where the graph reaches a high/low point.Endpoints such as A and E can never be local max/min.B and D are local minima.C is a local maximum.

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Absolute Maxima and MinimaOver the entire domain of the graph, these are values that correspond to the largest or smallest value.A is the absolute maximum.D is the absolute minimum.

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Example• There is a local minimum at The local minimum value is .• There is NOT a local maximum at Why?• There is an absolute minimum at . The absolute minimum value is .• There is no absolute maximum. Why?

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Example• There is an absolute minimum at .• There is no absolute maximum.• There is not a local minimum.• There is not a local maximum.

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Exercise 48 – Page 2381. Local max?2. Local min?3. Absolute max?4. Absolute min?

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Exercise 49 – Page 2381. Local max?2. Local min?3. Absolute max?4. Absolute min?

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Exercise 54 – Page 2385 Use the calculator to find any local max or min between and 3.

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Exercise 57 – Page 238 Use the calculator to find any local max or min between and 4.

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Average Rate of ChangeTwo points on the graph: and .The average rate of change is simply the slope of the line between these 2 points:

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Secant LineThe line between 2 points on the graph is called the secant line.The equation is

The slope is the average rate of change.

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Exercise 67 – Page 239 a) Find the average rate of change from to 1.b) Find the equation of the secant line from to .

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Exercise 70 – Page 239 a) Find the average rate of change from to 3.b) Find the equation of the secant line from to .