🔍 Identifying a Quadratic → ax² + bx + c
- Highest power is x² (x-squared). Graph = PARABOLA (U-shape or upside-down U-shape).
- a > 0 (positive) → opens UP (min point at bottom) | a < 0 (negative) → opens DOWN (max at top)
🎯 VERTEX (Max or Min)
- Tip-top or bottom point
- Formula: x = −b / (2a)
- Then plug x in to get y!
- e.g. y = x²−4x+3:
- x = 4/(2·1) = 2
- y=4−8+3=−1 → (2,−1)
📍 X-INTERCEPTS (Roots)
- Where parabola hits x-axis
- Set y = 0 then FACTOR:
- x²−4x+3 = 0
- (x−1)(x−3) = 0
- x = 1 or x = 3
- Two x-intercepts!
📌 Y-INTERCEPT
- Where parabola hits y-axis
- Set x = 0:
- y = a(0)+b(0)+c
- y = c (last number!)
- e.g. y=3x²+5x+7:
- y-intercept = 7 → (0,7)
🧮 HOW MANY REAL ROOTS?
- Use DISCRIMINANT: b²−4ac
- If > 0 → TWO real roots
- If = 0 → ONE real root
- If < 0 → NO real roots
- (parabola misses x-axis)
- Check on Desmos graph!
📊 PARABOLA DIAGRAM
✂️ Factoring x² + bx + c
- Find 2 numbers that MULTIPLY to c AND ADD to b
- x² + 5x + 6: need ×=6 AND +=5 → 2 and 3 → (x+2)(x+3)
- x² − 4x + 3: need ×=3 AND +=−4 → −1 and −3 → (x−1)(x−3)
- Roots: x=1 and x=3
- Always CHECK by re-expanding using FOIL!
🔢 Desmos Strategies
3 Ways to Get the Answer!
Strategy 1: Evaluate / Find a Value
- Type with CAPITAL letters
- Type A*X^2 + B*X + C
- Add sliders A, B, C
- Set values to your problem
- See vertex coordinates
- Discriminant: check B²−4·A·C
Strategy 2: Graph / Visualize
- Type y = x^2-4*x+3
- Parabola appears instantly!
- VERTEX: click the bottom point
- X-INTERCEPTS: click where it
- crosses the x-axis (the roots!)
- Y-INTERCEPT: click y-axis crossing
Strategy 3: Generate Table
- Type y = x^2−4*x+3 → Table
- Find rows where y = 0
- Those x-values = ROOTS!
- Find minimum y = vertex!
- x=0 row → y-value = y-intercept!
- Table gives ALL key features!
Remember This!
- Y-intercept = the c value (last number). Just set x=0! Discriminant = b²−4ac (tells number of roots)
- Positive discriminant → 2 roots. Zero → 1 root. Negative → no real roots (parabola misses x-axis).
- Vertex x = −b/(2a). Plug back in to get vertex y. Vertex is the MAX or MIN of the parabola.