Terminology Increasing As your x finger moves right, your y
finger moves up. Decreasing As your x finger moves right, your y
finger moves down Constant As your x finger moves right, your y
finger does not move Because x is the independent variable,
increasing/decreasing/constant intervals are described in terms of
values of x
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Thinking with your fingers
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Interval Notation Increasing on (-,-2) Decreasing on (-2,2)
Increasing on (2,)
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Extrema and Local Extrema The maximum of a graph is its highest
y value. The minimum of a graph is its lowest y value. A local
maximum of a graph is a y value where the graph changes from
increasing to decreasing. (Stops getting bigger) A local minimum of
a graph is a y value where the graph changes from decreasing to
increasing (Stops getting smaller) These extrema are all y values.
But because x is the independent variable, they are often described
by their location (x value).
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Extrema No minimum, or minimum of - No maximum, or maximum of
Local maximum of (x)=5.3333 at x=-2 Local minimum of (x)=-5.3333 at
x=2
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Using your graphing utility, graph the function below and
determine on which x-interval the graph of h(x) is increasing:
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Piecewise Functions
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The Bank Problem Frances puts $50 in a bank account on Monday
morning every week. Draw a graph of what Frances's bank account
looks like over time. Put number of weeks on the horizontal axis,
and number of dollars in her account on the vertical axis.
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Frances Bank Account
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Writing a formula for Frances
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Piecewise function Function definition is given over interval
pieces Ex: Means: When x is between 0 and 2, use the formula 2x+1.
When x is between 2 and 5, use the formula (x-3) 2
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Example Find (3) Check the first condition: 03
Consider the piecewise function below: Find h(3). Check first
condition: 3>5? FALSE Check second condition: 35? TRUE Use x-5
3-5=-2 B) -2
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Combining Functions (Algebra of Functions)
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Things to remember Function notation (x)=2x-1 is a function
definition x is a number (x) is a number 2x-1 is a number is the
action taken to get from x to (x) Multiply by 2 and add -1
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Things to remember Function notation (x)=2x-1 is a function
definition 3 is a number (3) is a number 2*3-1 is a number (its 5)
is the action taken to get from 3 to (3) Multiply by 2 and add
-1
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In visual form
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Numbers can be added and multiplied
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When I add and multiply the results of functions, I create a
new function x f(x) g(x) f(x)+g(x) + f g
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When I add and multiply the results of functions, I create a
new function x f(x) g(x) f(x)+g(x) + f g
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I can give this new function a name xf(x)+g(x) f+g
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I can give this new function a name xf(x)+g(x) f+g The function
f+g is the action: 1)Do f to x. get f(x) 2)Do g to x. get g(x)
3)Add f(x)+g(x)
WARNING (fg)(x) and f(g(x)) are not the same thing (fg)(x)
means do f to x, then do g to x, then multiply the numbers f(x) and
g(x). f(g(x)) means do g to x, get the number g(x), then do f to
the number g(x) No multiplying. We will talk more about f(g(x))
next time.
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Example Find (g/f)(-2)
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Graphing sums Function has a graph. Function g has a graph
Function (+g) also has a graph. Can I find the graph of (+g) from
the graphs of and g? Hint: the answer is yes.
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How to sum function graphs
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What if one of them is negative?
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Use as many points as you need
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Given the function definitions below: Find (+g)(3). a)-11 b)3
c)0 d)-21 e)None of the above
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Given the function definitions below: Find (+g)(3).
f(3)=2*3-1=5 g(3)=4-3 2 =-5 (f+g)(3)=f(3)+g(3)=5+-5=0 c) 0
