1. Functions – Transformations, Classification, Combination
Calculus P-3b

2. Functions - Graphing Vertical Line Test Identify 8 Basic Functions
f(x)=x
f(x)=x2
f(x)=x3
f(x)=x ½
f(x)=abs(x)
f(x)=1/x
f(x)=sin x
f(x)=cosx
Functions - Graphing

3. The squaring function

4. The square root function

5. The absolute value function

6. The cubing function

7. The cube root function

8. Rational Function

9. Sine Function

10. Cosine Function

11. Vertical Translation Vertical Translation For b > 0,
the graph of y = f(x) + b is the graph of y = f(x) shifted up b units;
the graph of y = f(x)  b is the graph of y = f(x) shifted down b units.
Vertical Translation

12. Horizontal Translation
For d > 0,
the graph of y = f(x  d) is the graph of y = f(x) shifted right d units;
the graph of y = f(x + d) is the graph of y = f(x) shifted left d units.
Horizontal Translation

13. Shifts Vertical shifts Horizontal shifts Moves the graph up or down
Impacts only the “y” values of the function
No changes are made to the “x” values
Horizontal shifts
Moves the graph left or right
Impacts only the “x” values of the function
No changes are made to the “y” values
Shifts

14. The values that translate the graph of a function will occur as a number added or subtracted either inside or outside a function. Numbers added or subtracted inside translate left or right, while numbers added or subtracted outside translate up or down.

15. Combining a vertical & horizontal shift
Example of function that is shifted down 4 units and right 6 units from the original function.
Combining a vertical & horizontal shift

16. The graph of f(x) is the reflection of the graph of f(x) across the x-axis.
The graph of f(x) is the reflection of the graph of f(x) across the y-axis.
If a point (x, y) is on the graph of f(x), then
(x, y) is on the graph of f(x), and
(x, y) is on the graph of f(x).
Reflections

17. Reflecting Across x-axis (y becomes negative, -f(x))
Across y-axis (x becomes negative, f(-x))
Reflecting

18. Sequence of transformations
Follow order of operations.
Select two points (or more) from the original function and move that point one step at a time.
f(x) contains (-1,-1), (0,0), (1,1)
1st transformation would be (x+2), which moves the function left 2 units (subtract 2 from each x),
pts. are now (-3,-1), (-2,0), (-1,1)
2nd transformation would be 3 times all the y’s,
pts. are now (-3,-3), (-2,0), (-1,3)
3rd transformation would be subtract 1 from all y’s, pts. are now (-3,-4), (-2,-1), (-1,2)
Sequence of transformations

19. Transformations

20. Polynomial Degree – Highest degree of any term in polynomial
Coefficient and degree of leading coefficient determines polynomial graph at end points
Polynomial Functions

21. Even degrees – End points both go same direction
Even degrees – End points both go same direction. Up if coefficient is positive, down if coefficient is negative
Odd degrees – End points go opposite directions. Up on right if coefficient is positive, down on right if coefficient is negative
Polynomial Functions

22. Discontinuity exists when denominators are zero
Rational Functions

23. Algebraic Functions – Functions that can be expressed as a finite number of operations.

24. Algebraic Functions

25. Algebraic Functions - Practice
Find:
Algebraic Functions - Practice

26. Solve this problem, from the inside parenthesis out.
Composite Functions

27. Practice

28. Practice

29. Composite Function - Practice
Find: f(g(x))
and
g(f(x))
Composite Function - Practice

30. Even and Odd Functions

31. Even and Odd Functions Even Functions f(-x)=f(x)
Graph is symmetric wrt y-axis
Odd Functions
f(-x)=-f(x)
Graph is symmetric wrt origin
Even and Odd Functions

32. Example

33. Even and Odd Functions Odd (0,0), (-1,0),(1,0)
Determine if the following are even or odd, and then find the zeros of the functions
Odd
(0,0), (-1,0),(1,0)
Even and Odd Functions

34. Even and Odd Functions Odd (0,0)
Determine if the following are even or odd, and then find the zeros of the functions
Odd
(0,0)
Even and Odd Functions

35. Right 3
Left 3
Down 3
Practice Problems

36. Practice Problems Up 3 Reflect across x-axis Reflect across y-axis
Stretch by 3 in y direction
Practice Problems

37. Practice Problems

38. Exploration

39. pgs 27 – 29
31, 33, 37, 39-42, 49, 51, 52, 53, 55, 59, 60, 72
Homework