Think VENDING MACHINE!
- DOMAIN = buttons you press (inputs / x-values) — look LEFT & RIGHT on the graph
- RANGE = snacks that come out (outputs / y-values) — look UP & DOWN on the graph
- Interval notation: [a,b] includes endpoints | (a,b) excludes endpoints | use ∞ for "goes forever"
📏 LINEAR Function y = mx + b
- Graph = STRAIGHT LINE — goes forever both ways!
- Domain: ALL real numbers (−∞, ∞) ← ALWAYS
- Range: ALL real numbers (−∞, ∞) ← ALWAYS
- EXCEPTION: horizontal line y=3 → Range = {3} only!
- EXCEPTION: vertical line x=3 → NOT a function!
🎯 QUADRATIC Function y = ax²+bx+c
- Graph = PARABOLA (U or upside-down U)
- Domain: ALWAYS all real numbers (−∞, ∞)
- Range: depends on VERTEX y-value!
- Opens UP (a>0): Range = [vertex y, ∞)
- Opens DOWN (a<0): Range = (−∞, vertex y]
- e.g. y=x²−4: vertex (0,−4) → Range=[−4, ∞)
📊 Reading Domain & Range
- FROM A TABLE: Domain = list all x-values | Range = list all y-values
- e.g. x:{1,2,3,4,5} → Domain={1,2,3,4,5} y:{3,5,7,9,11} → Range={3,5,7,9,11}
- FROM A GRAPH: Domain = how far LEFT and RIGHT? | Range = how far UP and DOWN?
- Open circle ○ = NOT included → use ( ) | Closed circle ● = IS included → use [ ] | Arrow = goes to ∞
🔢 Desmos Strategies
3 Ways to Get the Answer!
Strategy 1: Evaluate / Find a Value
- Type with CAPITAL letters
- For quadratic: A*X^2+B*X+C
- Add sliders A, B, C
- Drag to change shape
- Observe how range changes
- Positive A = opens up (Range up)
Strategy 2: Graph / Visualize
- Type y = x^2 - 4 (lowercase)
- Graph appears — observe it!
- Domain: how far left and right?
- → all the way both directions!
- Range: click the lowest point
- → that y-value starts the range!
Strategy 3: Generate Table
- Type y = x^2 - 4 → Table
- x: −3,−2,−1,0,1,2,3
- y: 5, 0,−3,−4,−3,0,5
- Smallest y in table = min Range
- Domain = all x values (no limit)
- Range = [min y, infinity)
Remember This!
- For BOTH linear and quadratic: Domain is ALMOST ALWAYS all real numbers (−∞, ∞)!
- Range for quadratic: find vertex y. Opens UP → y ≥ vertex y. Opens DOWN → y ≤ vertex y.
- [ ] = includes that number (closed dot on graph). ( ) = does NOT include it (open dot).