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Features of Graphs of FunctionsPowerPoint Presentation

Features of Graphs of Functions

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### Piecewise Functions determine on which

### Combining Functions determine on which

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

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

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:

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’ determine on which Bank Account

Writing a formula for Frances determine on which

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 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

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

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: determine on which Find h(3).

- 7
- -2
- -3
- Both (a) and (b)
- None of the above

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

(Algebra of Functions)

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 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 determine on which

Numbers can be added and multiplied determine on which

Numbers can be added and multiplied determine on which

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 function

f(x)

f

x

f(x)+g(x)

+

g

g(x)

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)

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

(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.

Example

- Find (g/f)(-2)

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 (ƒ+g)(3).

- -11
- 3
- 0
- -21
- None of the above

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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