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

?

months

$

Not Correct

weeks

$

Looking at the problem day by day

weeks

$

Solution

weeks

$

C)

Whirlwind Function Tour

Piecewise FunctionsAbsolute Value

Monomials (AKA Power Functions)

Writing a formula for Frances

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) 7B) -2C) -3D) 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

Monomials

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

x f(x)

-2 -16

-1 -2

0 0

1 2

2 16

Even Power MonomialsPower 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 MonomialsPower 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≥0Range: y≥0

1/3 Power

Domain: all real numbersRange: 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≠0Range: y>0

Domain: x≠0Range: 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) $6.75B) $5.50C) $4.50D) $3.50E) 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