The document discusses power functions and characteristics of polynomial functions. It defines power functions as functions of the form f(x)= xa where a is a fixed number. It also discusses key features of polynomial functions such as reflection, local/absolute maximum/minimum points, and the relationship between finite differences and the polynomial equation.

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

2. 1.1
Power 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. Definition
• Leading coefficient is the one with the
highest power.
- Coefficient must be a whole
number.
• Degree is the power of x.
• End Behaviour of 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
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
Power Function Degree Name
Y=a 0 Constant
values of a function for the
Y=ax 1 Linear
values of the variable. Quadratic
Y=ax
2 2
Y=ax3 3 Cubic
Y=ax4 4 Quartic
Y=ax5 5 Quintic
Power Functions

7. 1.2
Characteristics of
Polynomial Functions

8. 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
0 1 2 3
3.5 0
3 0.5
2.5 1
2 1.5
1.5 2
1 2.5
0.5 3
0 3.5
0 1 2 3
Key Features

11. Reflection in terms of End Behavior…
Q 3 to Q 1 becomes Q 2 to Q 4 (odd degree)
Q 2 to Q 1 becomes Q 3 to Q 4 (even degree)
Key Features

12. 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

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

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

15. COMPARISON between odd and
even degree function
Key feature Odd degree Even degree
No. of absolute 0 0 1 1
max/min
points.
Total number of Maximum (n -1) Maximum (n-1) Maximum (n-1) Maximum (n-1)
local max/min
points.
No. of absolute Maximum (n -1) Maximum (n -1) Maximum (n -1) Maximum (n -1)
max/min
points.
Key Features

16. Finite Differences

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

18. For a polynomial function of degree
n, where n is a positive integer, the nth
differences.
Are equal(or constant)
Have equal to a[n*(n-1)*…*2*1]
,where a is the leading coefficient.
Key Features

19. Differences
x y 1st differences 2nd differences 3rd differences
-3 -36
-2 -12 -12-(-36)=24
-1 -2 -2-(-12)=10 10-24=-14
0 0 0-(-2)=2 2-10=-8 -8-(-14)=6
1 0 0-0=0 0-2=-2 -2-(-8)=6
2 4 4-0=4 4-0=4 4-(-2)=6
3 18 18-4=14 14-4=10 10-4=6
4 48 48-18=30 30-14=16 16-10=6
Key Features

20. • 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 coefficientis positive,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