Each part of graph is described as: 1)Increasing : function values increase from left to right 2)Decreasing: function values decrease 3)Constant function values are constant Each part of graph is described as: 1)Increasing : function values increase from left to right 2)Decreasing: function values decrease 3)Constant function values are constant 1. Definition: Increasing/Decreasing/Constant

Use x-intervals to describe this property! 1)Increasing 2)Decreasing 3)Constant Use x-intervals to describe this property! 1)Increasing 2)Decreasing 3)Constant 2. Determine the intervals of Increasing/Decreasing/Constant

3. Definition: Local Maxima and Minima local maximum:“peaks” or “high points” on graph ( is NOT a max!) local minimum: “valleys” or “low points” on graph Local maximum at x = ___ Value of local maximum is f(a) Local maxima are the points: Local minimum at x = Local minima at: Local minimum at x = Local minima at:

Local Minima and Maxima List intervals of increasing/decreasing and all maxima/minima. p. 89 #12.

4 a) Using the Graphing Calculator Use a graphing utility 1.Approximate all local maxima or minima. 2. Find all intervals where f is increasing and decreasing. To find max/min on calculator 1.y1= 2.2 nd TRACE (CALC) 3.3:minimum or (4:maximum) 4.move left of min or max ENTER 5.move right of min or max ENTER 6.ENTER To find max/min on calculator 1.y1= 2.2 nd TRACE (CALC) 3.3:minimum or (4:maximum) 4.move left of min or max ENTER 5.move right of min or max ENTER 6.ENTER A function changes from increasing to decreasing at a maximum A function changes from decreasing to increasing at a minimum A function changes from increasing to decreasing at a maximum A function changes from decreasing to increasing at a minimum

5. Symmetry Some functions have graphs with symmetry: Even function symmetry with respect to y-axis ODD function symmetry with respect to origin no symmetry

5) Even and Odd Functions To show algebraically: All EVEN functions satisfy To show algebraically: All ODD functions satisfy Even function symmetry with respect to y-axis Odd function symmetry with respect to origin

5 b) Examples. Determine if even/odd Determine whether the following functions are even, odd, or neither. Then state any symmetry (origin/y-axis) 1) 2) 3) 4)

5) Even and Odd Functions Is the same as ? Evaluate and simplify: Evaluate : Then is an Even function Is the same as ? Then is an ODD function is neither and no symmetry YES NO YES NO