Notes: Non-Linear Modeling

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AM5 – Non-Linear
Modelling

Checklist
1) Quadratic Functions (Parabolas)
2) Cubic Functions
3) Exponential Functions
4) Hyperbolic Functions
5) Direct Variation
6) Indirect Variation

1) Quadratic Functions
 A quadratic function is an equation that contains x2 as the

highest power.
 Examples are:

y = x2
 y = x2 + 1


y = x2 + 3x - 1
 y = -2x2 + 4x
 y = x2 – 9


Example 1


Draw the graph of y = x2 + 1

NOTE: The special features of the parabola include:
 The vertex
 The concavity

Example 2


The area (A) of a rectangular garden of length x metres is given by A = 6x − x2.
a) Complete the following table of values:

b) Draw the graph of A = 6x − x2 using the table of ordered pairs.
c) What is the maximum area of the garden?
d) What is the maximum area of the garden?

Example 2 - Answers
a) Complete the following table of values:

Homework

Ex 12A p. 353 – 354 Q1 – 14

2) Cubic Functions
 A cubic function is an equation that contains x3 as the highest

power.
 Examples are:

y = x3
 y = x3 + 1


y = x3 – x2 + 3x - 1
 y = -2x3 + 4x
 y = x3 – 9


Example 1


Draw the graph of y = x3

NOTE: The special features of the cubic include:
 The point of inflexion
 The gradient

3) Exponential Functions
 An exponential function is an equation that contains x as the

exponent (power).
 Examples are:

y = 2x
 y = 2x + 1


y = 32x
 y = - 2x
 y = 3-x


Example 1


Draw the graph of y = 3x x y

-2

-1

0

1

2

Example 1
NOTE: The special features of the exponential include:
 The exponential has a “tail” and it increases very quickly

4) Hyperbolic Functions
 A hyperbolic function is an equation that contains x on the a y denominator of a fraction i.e. where a is a constant. x  Examples are:

1 y x
- y  3 x 4 y 3x

Example 1


Draw the graph of y x y

-2

1 x -1

0

1

2

Example 1


Draw the graph of y 

1 x NOTE: The special features of the hyperbola include:
 The asymptotes
 The branches

Example 2


Draw the graph of y  x y

-2

2 x -1

0

1

2

Example 2


Draw the graph of y 

2 x NOTE: The special features of the hyperbola include:
 The asymptotes
 The branches

Homework

Ex 12B p. 359 – 360 ALL
QUESTIONS
Due Tuesday

Exponential Growth and Decay
HSC 2008 Q25

Exponential Growth and Decay
HSC 2004 Q26

Homework

Ex 12C p. 364 – 365 ALL QUESTIONS
Due Friday
Your Cheat Sheets for AM4 and AM5 will be checked!

5) Direct Variation


This variation (also called direct proportionality) occurs when one variable follows another i.e. when one increases, so does the other. When one decreases, so does the other.



Direct variation problems can occur in the following ways:

a) y is directly proportional to x.

i.e. y x

b) y is directly proportional to the square of x.
c)

y is directly proportional to