advanced functions part i

Advanced

Functions E-PresentationPrepared by:

Tan Yu HangTai Tzu YingWendy Victoria VazTan Hong YeeVoon Khai SamWei Xin

Part I

1.1Power Functions

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

• 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

• 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 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

• 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 Function Degree Name

Y=a 0 Constant

Y=ax 1 Linear

Y=ax2 2 Quadratic

Y=ax3 3 Cubic

Y=ax4 4 Quartic

Y=ax5 5 Quintic

Power Functions

1.2 Characteristics of Polynomial Functions

Key Feature

s

Reflection (-/+ Sign)

End Behaviour

Number of local

maximum/minimu

m points

Number of absolute (global) maximum / minimum points

Number of

x-intercepts.

Key Features

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

Example: Reflection on

X-axis

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

3.5

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

3.5

Key Features

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

LOCAL maximum/minimum

pointLocal maximum or minimum point means the

largest or smallest value in a graph of a polynomial function within the GIVEN domain.

Key Features

ABSOLUTE maximum/minimum

pointThe largest or smallest point in a graph of polynomial function on its ENTIRE domain.

Key Features

Local max. point

Absolute min. point

Local min. point

Key Features

COMPARISON between odd and even degree function

Key feature Odd degree Even degree

No. of absolute max/min points.

0 0 1 1

Total number of local max/min points.

Maximum (n -1) Maximum (n-1) Maximum (n-1) Maximum (n-1)

No. of absolute max/min points.

Maximum (n -1) Maximum (n -1) Maximum (n -1) Maximum (n -1)

Key Features

Finite Differenc

esFinite Differences

What is the relationship

between finite

differences and the

equation of a polynomial function?

Finite Differences

FINITE DIFFERENCES

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

Example of finite difference table

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

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 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=6aa=1

Key Features

That’s all for Chapter 1.1 and 1.2,

Do stay tuned to Part II to know more

about 1.3! ;)