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Features of Graphs of Functions. Thinking with your fingers. 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

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Terminology
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



Interval notation
Interval Notation

Increasing on (2,∞)

Increasing on (-∞,-2)

Decreasing on (-2,2)


Extrema and local extrema
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).


Extrema
Extrema

No maximum, or maximum of ∞

Local maximum of ƒ(x)=5.3333

at x=-2

Local minimum of ƒ(x)=-5.3333

at x=2

No minimum, or minimum of -∞


Using your graphing utility, graph the function below and determine on which x-interval the graph of h(x) is increasing:


Using your graphing utility, graph the function below and determine on which x-interval the graph of h(x) is increasing:


Piecewise functions

Piecewise Functions determine on which


The bank problem
The Bank Problem determine on which

  • 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.


Frances bank account
Frances’ determine on which Bank Account


Writing a formula for frances
Writing a formula for Frances determine on which


Piecewise function
Piecewise function determine on which

  • 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”


Example
Example determine on which

  • Find ƒ(3)

  • Check the first condition: 0≤3<2? FALSE.

  • Check the second condition: 2≤3<5? TRUE.

  • Use (x-3)2

  • (3-3)2=02=0

  • ƒ(3)=0


Example1
Example determine on which

For the interval of x [0,2) draw a graph of 2x+1

For the interval of x [2,5) draw a graph of (x-3)2


Example2
Example determine on which

For the interval of x [0,2) draw a graph of 2x+1

For the interval of x [2,5) draw a graph of (x-3)2


Consider the piecewise function below find h 3
Consider the piecewise function below: determine on which Find h(3).

  • 7

  • -2

  • -3

  • Both (a) and (b)

  • None of the above


Consider the piecewise function below find h 31
Consider the piecewise function below: determine on which Find h(3).

  • Check first condition: 3>5? FALSE

  • Check second condition: 3≤5? TRUE

  • Use x-5

  • 3-5=-2

  • B) -2


Combining functions

Combining Functions determine on which

(Algebra of Functions)


Things to remember
Things to remember determine on which

  • Function notation

  • ƒ(x)=2x-1 is a function definition

  • x is a number

  • ƒ(x) is a number

  • 2x-1 is a number

  • ƒ is the action taken to get from x to ƒ(x)

    • Multiply by 2 and add -1


Things to remember1
Things to remember determine on which

  • Function notation

  • ƒ(x)=2x-1 is a function definition

  • 3 is a number

  • ƒ(3) is a number

  • 2*3-1 is a number (it’s 5)

  • ƒ is the action taken to get from 3 to ƒ(3)

    • Multiply by 2 and add -1


In visual form
In visual form determine on which




When i add and multiply the results of functions i create a new function
When I add and multiply the results of functions, I create a new function

f(x)

f

x

f(x)+g(x)

+

g

g(x)


When i add and multiply the results of functions i create a new function1
When I add and multiply the results of functions, I create a new function

f(x)

f

x

f(x)+g(x)

+

g

g(x)



I can give this new function a name1
I can give this new function a name

f+g

x

f(x)+g(x)

  • The function f+g is the action:

  • Do f to x. get f(x)

  • Do g to x. get g(x)

  • Add f(x)+g(x)


Example3
Example

2x

f

x

2x+3x

+

g

3x


Example4
Example

2x

f

x

5x

+

g

3x


Example5
Example

f+g

x

5x


Notation
Notation

  • (f+g)(x)=f(x)+g(x)

  • (f-g)(x)=f(x)-g(x)

  • (fg)(x)=f(x)*g(x)

  • (f/g)(x)=f(x)/g(x), g(x)≠0


Warning
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.


Example6
Example

  • Find (g/f)(-2)


Graphing sums
Graphing sums

  • Function ƒ has a graph.

  • Function g has a graph

  • Function (ƒ+g) also has a graph.

  • Can I find the graph of (ƒ+g) from the graphs of ƒ and g?

    • Hint: the answer is yes.















Given the function definitions below find g 3
Given the function definitions below: Find (ƒ+g)(3).

  • -11

  • 3

  • 0

  • -21

  • None of the above


Given the function definitions below find g 31
Given the function definitions below: Find (ƒ+g)(3).

  • f(3)=2*3-1=5

  • g(3)=4-32=-5

  • (f+g)(3)=f(3)+g(3)=5+-5=0

    c) 0


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