Linear & Quadratic Functions Quick Reference
Module 2: College Algebra
Linear Functions
m = (y₂ - y₁) / (x₂ - x₁)
| Form | Equation |
|---|---|
| Slope-Intercept | y = mx + b |
| Point-Slope | y - y₁ = m(x - x₁) |
| Standard | Ax + By = C |
Special Lines
Parallel: Same slope (m₁ = m₂)
Perpendicular: m₁ × m₂ = -1
Slope Interpretation
| Slope | Direction |
|---|---|
| m > 0 | Rising (left to right) |
| m < 0 | Falling (left to right) |
| m = 0 | Horizontal |
| Undefined | Vertical |
Linear Modeling
Form: y = mx + b
m = rate of change
b = starting value
Applications
- Cost: C = (unit price)(x) + fixed
- Temp: F = (9/5)C + 32
- Depreciation: V = mt + initial
Quadratic - Standard Form
Direction:
a > 0: Opens up (minimum)
a < 0: Opens down (maximum)
x = -b/(2a)
y = f(-b/(2a))
x = -b/(2a)
Quadratic - Vertex Form
Vertex: (h, k)
a: Opens up (a>0) or down (a<0)
h: Horizontal shift
k: Vertical shift
Solving Quadratics
Method 1: Factoring
Factor, set each = 0, solve
x = [-b ± √(b² - 4ac)] / (2a)
Method 3: Complete Square
1. x² + bx = -c
2. Add (b/2)² to both sides
3. Factor, take √, solve
Discriminant
| Value | Solutions |
|---|---|
| D > 0 | Two real |
| D = 0 | One real (repeated) |
| D < 0 | No real (complex) |
Finding Intercepts
| Type | Method |
|---|---|
| Y-intercept | Set x = 0 |
| X-intercepts | Set y = 0, solve |
Transformations
f(x) = a(x - h)² + k
|a| > 1: Narrower
0 < |a| < 1: Wider
a < 0: Reflect x-axis
h > 0: Right h units
h < 0: Left |h| units
k > 0: Up k units
k < 0: Down |k| units
Completing the Square Process
1. Group: (x² + bx) + c
2. Take (b/2)²: Add and subtract inside parentheses
3. Factor: (x + b/2)² - (b/2)² + c
4. Simplify: (x + b/2)² + [c - (b/2)²]
Key Strategies
- Use slope formula for two points
- Perpendicular slopes multiply to -1
- Vertex gives max/min value
- Check discriminant before solving
- Factor when possible (fastest)
- Use quadratic formula when factoring is hard
- Always verify solutions
- Graph to visualize
Common Mistakes to Avoid
- Confusing (x - h)² with (x + h)² in vertex form - watch signs!
- Forgetting ± in quadratic formula
- Sign errors when distributing negative numbers
- Mixing up perpendicular vs parallel slope rules
- Not simplifying fractions in final answers