1.5 Increasing/Decreasing; Max/min Tues Sept 16 Do Now Graph f(x) = x^2 - 9.

1.5 Increasing/Decreasing; Max/minTues Sept 16

Do NowGraph f(x) = x^2 - 9

HW Review: p.87 #43-51, 55-61

• 43) (-infinity, infinity)• 45) (-infinity, 0) U (0, infinity)• 47) (-infinity, 2) U (2, infinity)• 49) (-infinity, -1) U (-1, 5) U (5, infinity)• 51) (-infinity, 8]• 55) D: [0, 5] R: [0, 3]• 57) D: [-2pi, 2pi] R: [-1, 1]• 59) D: (-infinity, infinity) R: {-3}• 61) D: [-5, 3] R: [-2, 2]

Increasing and Decreasing Functions

• A function’s behavior can be described as one of three types:– Increasing– Decreasing– Constant

• These behaviors can be described in interval notation as well

Ex 1

• Graph on page 120

• Determine the intervals on which the function is increasing, decreasing, constant

Relative Extrema

• Certain functions can have relative extrema, a point on the graph where the function changes from increasing to decreasing, or vice versa

• These are called relative maxima or minima– Sometimes called local maxima or minima

Finding relative extrema using a calculator

• 1) Type the function in “Y=“• 2) Graph• 3) “2nd” -> “Calc”• 4) Min or Max• 5) Left bound: Select a point to the left of the

max/min• 6) Right bound: Select a point to the right of the

max/min• 7) Guess: hit enter

Ex:

• Find the relative extrema of

and determine when it is increasing or decreasing

You try

• Graph each function. Find any relative extrema, and determine when each function is increasing or decreasing

• 1)

• 2)

• 3)

Closure

• What are relative extrema? How can we find them?

• HW: p.127 #1-21 odds

1.5 Piecewise FunctionsWed Sept 17

• Do Now• Graph• Find the relative minimum, and determine

where the function increases / decreases

HW Review: p.127 #1-21

• 1) a: (-5,1) b: (3, 5) c: (1, 3)• 3) a: (-3, -1), (3, 5) b: (1, 3) c: (-5, -3)• 5) a: (-inf, -8) (-3, -2) b: (-8, -6) c: (-6, -3), (-2, inf)• 7) D: [-5, 5] R: [-3, 3]• 9) D: [-5, -1] U [1, 5] R: [-4, 6]• 11) D: (-inf, inf) R: (-inf, 3]• 13) max: (2.5, 3.25), inc (-inf, 2.5) dec (2.5, inf)• 15) max: (-0.667,2.37), min: (0,2)

inc (-inf, -0.667) U (2, inf) dec (-0.667, 2)

17-21

• 17) min (0,0) inc (0, inf) dec (-inf, 0)• 19) max (0, 5) inc (-inf, 0) dec (0, inf)• 21) min (3, 1) inc (3, inf) dec (-inf, 3)

Piecewise functions

• A piecewise function is a function that uses different output formulas for different parts of the domain

• Each piece is only considered for the given domain

Graphing Piecewise Functions

• It is important to graph the endpoints of each piece, so we know where they fit in

Ex

• Graph

Ex 2

• Graph

Ex 3

• Graph

Greatest Integer Functions

• The greatest integer functionis defined as the greatest integer less than

or equal to x

This function is also known as a step function- Its graph looks like steps

Closure

• Graph

• HW: p.131 #39-49 odds, 59-63 odds