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

HW Review: p.87 #43-51, ) (-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) “2 nd ” -> “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 Functions Wed 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, ) U (2, inf)dec (-0.667, 2)

) 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 function is 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, odds