Frances puts \$50 in a bank account on Monday morning every week.

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

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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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

?

\$

months

Solution

C)‏

\$

weeks

Piecewise Functions

Absolute Value

Monomials (AKA Power Functions)‏

Whirlwind Function Tour
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).
• A) 7
• B) -2
• C) -3
• D) Both (a) and (b)‏
• E) None of the above
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

“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
• 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”)‏
|x|
• Also written abs(x)‏
• Pronounced “The absolute value of x”
• Definition
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
• If you want to know how a monomial behaves, you can ALWAYS make a table.
• f(x)=2x3
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
• 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)‏
• 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.
½ Power

Domain: x≥0

Range: y≥0

1/3 Power

Domain: all real numbers

Range: all real numbers

Negative Power
• Power of -1, -2, -3.572, etc
• Have a “hole” in the domain at x=0.
• Can't divide by zero!
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
• Power of any real number: 2/3, 8/5, 3.14159265, etc.
• Uses combinations of the behaviors above
• Can be tricky
• Make a table!

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
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