features of graphs of functions. thinking with your fingers
TRANSCRIPT
- Slide 1
- Features of Graphs of Functions
- Slide 2
- Thinking with your fingers
- Slide 3
- 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
- Slide 4
- Thinking with your fingers
- Slide 5
- Interval Notation Increasing on (-,-2) Decreasing on (-2,2) Increasing on (2,)
- Slide 6
- 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).
- Slide 7
- 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
- Slide 8
- 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
- Slide 11
- 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.
- Slide 12
- Frances Bank Account
- Slide 13
- Writing a formula for Frances
- Slide 14
- 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
- Slide 15
- 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
- Slide 20
- Combining Functions (Algebra of Functions)
- Slide 21
- 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
- Slide 22
- 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
- Slide 23
- In visual form
- Slide 24
- Numbers can be added and multiplied
- Slide 25
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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
- Slide 27
- 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
- Slide 28
- I can give this new function a name xf(x)+g(x) f+g
- Slide 29
- 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)
- Slide 30
- Example x 2x 3x 2x+3x + f g
- Slide 31
- Example x 2x 3x 5x + f g
- Slide 32
- Example x5x f+g
- Slide 33
- 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
- Slide 34
- 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.
- Slide 35
- Example Find (g/f)(-2)
- Slide 36
- 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.
- Slide 37
- 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
- Slide 50
- Given the function definitions below: Find (+g)(3). a)-11 b)3 c)0 d)-21 e)None of the above
- Slide 51
- 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