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Frances puts $50 in a bank account on Monday morning every week. - PowerPoint PPT Presentation


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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. A) ‏. B) ‏. C) ‏. D) ‏. E) ‏ None of the Above. ?. $.

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slide1

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.

slide2

A)‏

B)‏

C)‏

D)‏

E)‏

None of the Above

slide3
?

$

months

solution
Solution

C)‏

$

weeks

piecewise functions absolute value monomials aka power functions
Piecewise Functions

Absolute Value

Monomials (AKA Power Functions)‏

Whirlwind Function Tour
piecewise function
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”
consider the piecewise function below find f 3
Consider the piecewise function below: Find f(3).
  • A) 7
  • B) -2
  • C) -3
  • D) Both (a) and (b)‏
  • E) None of the above
solution1
Solution
  • Find f(3)‏
  • When x is less than or equal to 5, we use the formula “x-5”
  • Three is less than or equal to five.
  • 3-5=-2
  • B) -2
graphing
Graphing

“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”

special piecewise functions
Special Piecewise Functions
  • floor(x) (pronounced “floor of x”)‏
  • Writes functions like Frances' easily
  • Used a lot in computer programming
  • Not covered in this class
  • Ask me if you're curious
  • abs(x) (pronounced “the absolute value of x”)‏
slide14
|x|
  • Also written abs(x)‏
  • Pronounced “The absolute value of x”
  • Definition
monomial power function
Monomial (Power Function)‏
  • Any function of the form: f(x)=axb
  • (These are not exponentials: f(x)=abx)‏
  • Monomials have five sub families:
    • Even power
    • Odd power
    • 1/n Power
    • Negative Power
    • Real Power
making a table
Making a Table
  • If you want to know how a monomial behaves, you can ALWAYS make a table.
  • f(x)=2x3
even power monomials
Even Power Monomials
  • Power of 0,2,4,6,8,etc.
  • y never changes sign
  • f(x)=3
  • f(x)=3x0 g(x)=x2 h(x)=-0.5x4
odd power monomials
Odd Power Monomials
  • Power of 1,3,5,7,etc.
  • y changes sign at x=0
  • f(x)=2x f(x)=x3 f(x)=-2x5
1 n power monomials root functions
1/n Power Monomials (Root functions)‏
  • Powers are fractions: 1/2, 1/3, 1/4, etc.
  • Powers of 1/n are called the n-th root.
  • These are inverses of even or odd power monomials.
power
½ Power

Domain: x≥0

Range: y≥0

1 3 power
1/3 Power

Domain: all real numbers

Range: all real numbers

negative power
Negative Power
  • Power of -1, -2, -3.572, etc
  • Have a “hole” in the domain at x=0.
  • Can't divide by zero!
negative power1
Negative Power

Domain: x≠0

Range: y>0

Domain: x≠0

Range: y≠0

Notice that the Even/Odd power rules about y changing sign still work

real power
Real Power
  • Power of any real number: 2/3, 8/5, 3.14159265, etc.
  • Uses combinations of the behaviors above
  • Can be tricky
  • Make a table!
slide26

The average cost, in dollars, of producing the x photo albums is given by: Here, x is the number of photo albums. What is the average cost of producing 500 photo albums?

  • A) $6.75
  • B) $5.50
  • C) $4.50
  • D) $3.50
  • E) None of the above
solution2
Solution
  • A(x)=4.5+1000/x
  • A(500)‏
  • 4.5+1000/500=4.5+2=6.5
  • $6.50
  • E) None of the Above