Types of Functions

Type 1: Constant Function

f(x) = c

Example:

f(x) = 1

Type 2: Power Function

f(x) = xa

If a is a (+) integer

f(x) = xn where n = 1,2,3,4,5…..

-Shape depends on if n is even or odd

-As n increases the graph becomes flatter near 0 and steeper where x ≥ 1

If a is -1

f(x) = x-1 = 1/x

Hyperbola

If a = 1/n

Root Function

f(x) = x1/n = n√(x)

Polynomial

f(x) = axn + bxn – 1 + cx n – 2 ……..

Degree (n) – highest exponent value

1st Degree: f(x) = ax + b

2nd Degree: Quadratic: f(x) = ax2+ bx + c

Parabola

Higher Degrees

Type 3: Algebraic Functions

Can be constructed using algebraic operations (add, subtract, multiplication, division, square root)

f(x) = √(x2 + 1) f(x) = x4 – 16x2 + (x-2)3√(x)

x + √(x)

Shapes vary

Type 4: Trigonometric Functions

Tan(x) = sin(x)/ cos(x)

Type 5: Exponential Functions

f(x) = ax

Type 6: Log Function

f(x) = logax Inverse exponential

Related Functions

By applying certain transformations to graphs of given functions, we can obtain the graphs of related functions

Translations - Shifts

Vertical shifts y = f(x) + c

shifts c units up y = f(x) – c

shifts c units down

Horizontal shifts

y = f(x – c)

shifts right c units

y = f(x + c)

shifts left c units

Stretching and Compressing

y = cf(x)

stretched vertically by a factor of c

y = 1/c f(x)

compressed vertically by a factor of c

y = f(cx)

compressed horizontally by a factor of c

y = f(x/c)

strectched horizontally by a factor of c

Reflecting

y = -f(x)

graph reflects about the x-axis

y = f(-x)

graph reflects about the y-axis

Examples

Given y = √(x), sketch

a) y = √(x) - 2

b) y = √(x - 2)

c) y = - √(x)

d) y = 2√(x)

e) y = √(-x)