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

1. 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.

2. A)‏ B)‏ C)‏ D)‏ E)‏ None of the Above

3. ? $ months

4. Not Correct $ weeks

5. Looking at the problem day by day $ weeks

6. Solution C)‏ $ weeks

7. Piecewise Functions Absolute Value Monomials (AKA Power Functions)‏ Whirlwind Function Tour

8. Writing a formula for Frances

9. 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”

10. Consider the piecewise function below: Find f(3). • A) 7 • B) -2 • C) -3 • D) Both (a) and (b)‏ • E) None of the above

11. 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

12. 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”

13. 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”)‏

14. |x| • Also written abs(x)‏ • Pronounced “The absolute value of x” • Definition

15. Monomials

16. 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

17. Making a Table • If you want to know how a monomial behaves, you can ALWAYS make a table. • f(x)=2x3

18. 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

19. 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

20. 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.

21. ½ Power Domain: x≥0 Range: y≥0

22. 1/3 Power Domain: all real numbers Range: all real numbers

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

24. 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

25. 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!

26. 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

27. 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