1. Types of Functions

2. Type 1: Constant Function
f(x) = c
Example:
f(x) = 1

3. Type 2: Power Function
f(x) = xa

4. 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

5. If a is -1
f(x) = x-1 = 1/x
Hyperbola

6. If a = 1/n
Root Function
f(x) = x1/n = n√(x)

7. Polynomial f(x) = axn + bxn – 1 + cx n – 2 ……..
Degree (n) – highest exponent value
1st Degree: f(x) = ax + b

8. 2nd Degree: Quadratic: f(x) = ax2+ bx + c
Parabola

9. Higher Degrees

10. 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

11. Type 4: Trigonometric Functions

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

13. Type 5: Exponential Functions
f(x) = ax

14. Type 6: Log Function
f(x) = logax Inverse exponential

15. Related Functions
By applying certain transformations to graphs of given functions, we can obtain the graphs of related functions

16. Translations - Shifts Vertical shifts y = f(x) + c shifts c units up
shifts c units down

17. Horizontal shifts y = f(x – c) shifts right c units y = f(x + c)
shifts left c units

18. 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

19. Reflecting y = -f(x) graph reflects about the x-axis y = f(-x)
graph reflects about the y-axis

20. Examples Given y = √(x), sketch a) y = √(x) - 2 b) y = √(x - 2)
c) y = - √(x)
d) y = 2√(x)
e) y = √(-x)