College Algebra Quick Reference
Module 1: Functions & Coordinate Plane
KEY RULE: Each input (x) maps to EXACTLY ONE output (y) in a function!
Real Number Sets
| Set | Symbol | Example |
|---|---|---|
| Natural | ℕ | 1, 2, 3, ... |
| Whole | W | 0, 1, 2, ... |
| Integers | ℤ | ..., -1, 0, 1, ... |
| Rational | ℚ | 1/2, -3, 0.5 |
| Irrational | — | √2, π, e |
ℕ ⊂ W ⊂ ℤ ⊂ ℚ ⊂ ℝ
Quadrants
| Quadrant | Signs (x, y) |
|---|---|
| I | (+, +) |
| II | (−, +) |
| III | (−, −) |
| IV | (+, −) |
Essential Formulas
Distance:
d = √[(x₂-x₁)² + (y₂-y₁)²]
d = √[(x₂-x₁)² + (y₂-y₁)²]
Midpoint:
M = ((x₁+x₂)/2, (y₁+y₂)/2)
M = ((x₁+x₂)/2, (y₁+y₂)/2)
Function Definition
Function: Each input (x) has exactly ONE output (y)
{(1,2), (2,3), (3,4)}
{(1,2), (1,3), (2,4)}
Vertical Line Test
If ANY vertical line hits graph MORE THAN ONCE → NOT a function
Domain & Range
Domain: All possible x-values (inputs)
Range: All possible y-values (outputs)
Common Restrictions
• Division: g(x) ≠ 0
• Square root: expression ≥ 0
Function Arithmetic
(f + g)(x) = f(x) + g(x)
(f - g)(x) = f(x) - g(x)
(f · g)(x) = f(x) · g(x)
(f / g)(x) = f(x) / g(x), g(x) ≠ 0
(f - g)(x) = f(x) - g(x)
(f · g)(x) = f(x) · g(x)
(f / g)(x) = f(x) / g(x), g(x) ≠ 0
Parent Functions
| Name | Equation |
|---|---|
| Linear | f(x) = x |
| Quadratic | f(x) = x² |
| Cubic | f(x) = x³ |
| Absolute Value | f(x) = |x| |
| Square Root | f(x) = √x |
| Reciprocal | f(x) = 1/x |
Transformations
| Type | Form | Effect |
|---|---|---|
| Vertical Shift | f(x) + k | Up k (k>0) Down k (k<0) |
| Horizontal Shift | f(x - h) | Right h (h>0) Left h (h<0) |
| Vertical Stretch | a·f(x) | Stretch (|a|>1) Compress (0<|a|<1) |
| Reflect x-axis | -f(x) | Flip upside down |
| Reflect y-axis | f(-x) | Flip left-right |
HORIZONTAL SHIFTS
OPPOSITE OF SIGN!
f(x-3) → right 3
f(x+3) → left 3
OPPOSITE OF SIGN!
f(x-3) → right 3
f(x+3) → left 3
Transformation Order
- Horizontal shifts (inside)
- Stretches & reflections
- Vertical shifts (outside)
Example: Complete Transformation
g(x) = -2(x - 3)² + 1 from f(x) = x²
- Shift RIGHT 3: (x - 3)²
- Stretch by 2: 2(x - 3)²
- Reflect over x-axis: -2(x - 3)²
- Shift UP 1: -2(x - 3)² + 1
Result: Upside-down parabola, vertex at (3, 1)
Common Mistakes to Avoid
- Horizontal shift sign confusion
- Mixing f(x)+k and f(x+k)
- Wrong transformation order
- Forgetting domain restrictions
- Confusing -f(x) and f(-x)
- Vertical line test errors
Quick Examples
Distance
Points: (0,0) and (3,4)
d = √[9+16] = √25 = 5
Midpoint
Points: (2,3) and (8,7)
M = (5, 5)
Function?
y = x² → YES
Circle → NO
Domain
f(x) = 1/(x-2)
x ≠ 2
Free College Algebra Platform • Safaa Dabagh • sdabagh.github.io • © 2025