1. Advanced Functions E-Presentation Prepared by: Part I Tan Yu Hang Tai Tzu Ying Wendy Victoria Vaz Tan Hong Yee VoonKhai Sam Wei Xin

2. 1.1Power Functions

3. The function f(x)=xa, where a is a fixed number. Usually only real values of xa are considered for real values of the base x and exponent a. The function has real values for all x > 0. If a is a rational number with an odd denominator, the function also has real values for all x < 0. Power Functions

4. If a is a rational number with an even denominator or if a is irrational, then xa has no real values for any x < 0. When x = 0, the power function is equal to 0 for all a > 0 and is undefined for a < 0; 00 has no definite meaning. Power Functions

5. Leading coefficient is the one with the highest power. - Coefficient must be a whole number. Degree is the power of x. End Behaviourof a graph function is the behavior of the y-values as x increases (that is, as x approaches to positive infinity it is written as x and as x decreases (as x approaches to negative infinity it is written as x) Line of Symmetry is the line which divides the graph into two parts and it reflects of the other in the line x=a Definition Power Functions

6. Point symmetry is a point (a,b) if each part of the graph on one side of (a,b) can be rotated 180 degrees to coincide with part of the graph on the other side of (a,b) Range is the set of all possible values of a function for the values of the variable. Power Functions

7. 1.2Characteristics of Polynomial Functions

8. Reflection (-/+ Sign) Number of x-intercepts. End Behaviour Key Features Number of absolute (global) maximum / minimum points Number of local maximum/minimum points Key Features

9. Reflection The relationship between f(x) and-f(x) is reflection in y-axis. The relationship between f(x) and f(-x) is reflection in x-axis. Key Features

10. Example: Reflection on X-axis Key Features

12. Q 2 to Q 1 becomes Q 3 to Q 4 (even degree)Key Features

13. LOCAL maximum/minimum point Local maximum or minimum point means the largest or smallest value in a graph of a polynomial function within the GIVEN domain. Key Features

14. ABSOLUTE maximum/minimum point The largest or smallest point in a graph of polynomial function on its ENTIRE domain. Key Features

15. Local max. point Local min. point Absolute min. point Key Features

16. COMPARISON between odd and even degree function Key Features

17. Finite Differences Finite Differences

18. What is the relationship between finite differences and the equation of a polynomial function? Finite Differences

20. Have equal to a[n*(n-1)*…*2*1] ,wherea is the leading coefficient.Key Features

21. Example of finite difference table Key Features

22. Explanation The 3rd differences are constant. The table of values represents a cubic function. The degree of the function is 3. From the table, the sign leading coefficientispositive,since 6 is positive. The value of the leading coefficient is the value of a such that 6=a[a*(n-1)*…*2*1]. Substitute n=3: 6=a(3*2*1) 6=6a a=1 Key Features

23. That’s all for Chapter 1.1 and 1.2, Do stay tuned to Part II to know more about 1.3! ;)