types of functions. type 1: constant function f(x) = c example: f(x) = 1
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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)