K McMullen 2012

Maths MethodsTypes of Functions 2

K McMullen 2012

Types Of Functions

Graphs of f(x)=xn where n is an even positive integer

The graphs of even functions are symmetrical about the y-axis

The maximal domain of the power function is R and the range is [0, ∞)

A function is even if f(-x)=f(x) for all values of x in the domain of f

Many-to-one function; every y value has a corresponding –x and x value

K McMullen 2012

Types Of Functions

The power form of a quadratic functionf(x)=a(x-b)2+c

a is the dilation factor from the x-axis

The turning point is at (b,c)

The equation of the axis of symmetry is the line x=b

The domain is R

The range is [c, ∞) if a>0 or (-∞,c] if a<0

b is the translation along the x-axis

c is the translation along the y-axis

K McMullen 2012

Types Of Functions

The power form of a quartic functionf(x)=a(x-b)4+c

a is the dilation factor from the x-axis

The turning point is at (b,c)

The equation of the axis of symmetry is the line x=b

The domain is R

The range is [c, ∞) if a>0 or (-∞,c] if a<0

b is the translation along the x-axis

c is the translation along the y-axis

K McMullen 2012

Types Of Functions

Graphs of f(x)=xn where n is an odd positive integer

As n increases, the branches of the curve move closer to the y-axis

The maximal domain of the function is R and the range is R

A function is odd if f(-x)=-f(x) for all values of x in the domain of x

K McMullen 2012

Types Of Functions

The power form of the cubic functiona is the dilation factor from the x-axis

(b,c) is the stationary point of inflection

The domain is R

The range is R

b is the translation along the x-axis

c is the translation along the y-axis

K McMullen 2012

Types of Functions

Graphs of f(x)=1/xn where n is a positive odd integer

Maximal domain is R\{0}

Range is R\{0}

One-to-one and odd functions

Asymptotes at x=0 and y=0

K McMullen 2012

Types of Functions

Graphs of f(x)=1/xn where n is a positive even integer

Maximal domain is R\{0}

Range is R+

Many-to-one and even functions

Asymptotes at x=0 and y=0

K McMullen 2012

Types of Functions

Graphs of functions with fractional powersf(x)=xp/q

p and q are both positive odd integers with highest common factor of 1

Domain is R

Range is R

K McMullen 2012

Types of Functions

Graphs of functions with fractional powersf(x)=xp/q

p is a positive even integer and q is a positive odd integer with highest common factor of 1

Domain is R

Range is R+∪{0}

K McMullen 2012

Types of Functions

Graphs of functions with fractional powersf(x)=xp/q

p is a positive odd integer and q is a positive even integer with highest common factor of 1

Domain is R+∪{0}

Range is R+∪{0}

K McMullen 2012

Types of Functions

Graphs of functions with negative fractional powers

f(x)=x-p/q

p and q are both positive odd integers with highest common factor of 1

Domain is R+∪{0}

Range is R+∪{0}

Asymptotes at x=0 and y=0

K McMullen 2012

Types of Functions

Graphs of functions with negative fractional powers

f(x)=x-p/q

p is a positive even integer and q is a positive odd integer with highest common factor of 1

Domain is R+\{0}

Range is R+

Asymptotes at x=0 and y=0

K McMullen 2012

Types of Functions

Graphs of functions with negative fractional powers

f(x)=x-p/q

p is a positive odd integer and q is a positive even integer with highest common factor of 1

Domain is R+

Range is R+

Asymptotes at x=0 and y=0