advanced functions part i
TRANSCRIPT
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! ;)