🤔 Is it a FUNCTION? → The VERTICAL LINE TEST
- A function: every x-value has EXACTLY ONE y-value (one input → one output, no splitting!)
- TEST: draw a vertical line on the graph. Hits curve MORE than once? → NOT a function!
- Mapping: each input arrow points to EXACTLY ONE output. One input, two outputs = NOT a function!
📏 LINEAR y = mx + b
- Highest power: x¹
- Graph: STRAIGHT LINE
- Constant rate of change
- Equal x-changes = equal y-changes
- e.g. y = 3x + 2
- Shape: / or \ or flat line
🎯 QUADRATIC y = ax²+bx+c
- Highest power: x²
- Graph: PARABOLA (U or ∩)
- Has a vertex (max or min point)
- Rate of change increases/decreases
- e.g. y = x² − 4
- Shape: U or upside-down U
📈 EXPONENTIAL y = a·bˣ
- Variable is in the EXPONENT
- Graph: J-curve (rapid growth!)
- Multiplies by constant ratio
- b>1 = growth, 0
- e.g. y = 2ˣ or y = 3·(2)ˣ
- Shape: J-shaped curve
LINEAR — Straight Line
QUADRATIC — U-Shape
EXPONENTIAL — J-Curve
| Feature | LINEAR | QUADRATIC | EXPONENTIAL |
|---|---|---|---|
| Equation form | y = mx + b | y = ax²+bx+c | y = a · bˣ |
| Graph shape | Straight line | Parabola (U/∩) | J-curve |
| Has vertex? | No | YES! | No |
| Grows how? | Constant (add) | Increasing rate | Multiplies each step |
🔢 Desmos Strategies
3 Ways to Get the Answer!
Strategy 1: Evaluate / Find a Value
- Type A*X + B (linear, CAPS)
- Or A*X^2+B*X+C (quadratic)
- Or A*B^X (exponential)
- Add sliders, set values
- Find specific output for any input
- Compare outputs of all 3 types!
Strategy 2: Graph / Visualize
- Type all 3 to COMPARE shapes:
- y = 2*x + 1 (straight line)
- y = x^2 + 1 (U-shape)
- y = 2^x (J-curve)
- SAME GRAPH = equivalent functions!
- Identify by shape on graph!
Strategy 3: Generate Table
- Type each → click Table
- LINEAR: y-changes are EQUAL each step
- QUAD: y-changes INCREASE each step
- EXP: y-values DOUBLE each step
- Table pattern tells you the type!
- Is it a function? Each x = one y?
Remember This!
- Vertical Line Test: vertical line hits graph more than once = NOT a function. Simple!
- Linear = straight line, Quadratic = U-shape parabola, Exponential = J-curve.
- In exponential y=abˣ: a = starting value (y-intercept), b>1 = grows, 0