You know the formula. But the quadratic equation is more than just plugging in numbers — it's a complete picture of a parabola: where it crosses the x-axis, where it peaks, and how it's shaped.
This guide connects the algebra to the geometry.
The Full Picture: What ax² + bx + c Tells You
| Feature | Formula | What It Means | |---------|---------|---------------| | Roots | x = (−b ± √Δ) / 2a | Where parabola crosses x-axis | | Vertex | (−b/2a, f(−b/2a)) | Peak or valley of parabola | | Axis of Symmetry | x = −b/2a | Vertical line through vertex | | Y-intercept | (0, c) | Where parabola crosses y-axis | | Direction | Sign of a | a > 0 = opens up, a < 0 = opens down |
Discriminant Deep Dive
The discriminant Δ = b² − 4ac doesn't just count roots — it describes the parabola's relationship with the x-axis:
Δ > 0: Two Distinct Real Roots
The parabola punches through the x-axis at two points.
Example: x² − 5x + 6 = 0 → Δ = 25 − 24 = 1
Roots at x = 2 and x = 3. The parabola dips below the x-axis between these points.
Δ = 0: One Repeated Root
The parabola just kisses the x-axis at the vertex.
Example: x² − 6x + 9 = 0 → Δ = 0
Root at x = 3. The vertex sits exactly on the x-axis.
Δ < 0: No Real Roots
The parabola floats entirely above or below the x-axis.
Example: x² + 4 = 0 → Δ = −16
No real solutions. The parabola sits above the x-axis (minimum at y = 4).
Graphing a Parabola from the Equation
To sketch any parabola from ax² + bx + c, you need just 4 things:
- Direction: Check sign of a
- Y-intercept: Plot (0, c)
- Vertex: Calculate (−b/2a, f(−b/2a))
- X-intercepts: Solve using the quadratic formula
Worked Example: f(x) = −2x² + 8x − 5
- a = −2 (opens downward — maximum)
- Y-intercept: (0, −5)
- Vertex: x = −8/(2·(−2)) = 2, y = −2(4) + 8(2) − 5 = 3 → (2, 3)
- Roots: Δ = 64 − 40 = 24 → x = (−8 ± √24)/(−4) = (8 ± 2√6)/4 → x ≈ 0.78 and x ≈ 3.22
The parabola opens downward, peaks at (2, 3), crosses the x-axis at ≈0.78 and ≈3.22, and crosses the y-axis at −5.
The Axis of Symmetry
Every parabola is symmetric about the vertical line x = −b/2a.
This means: if (p, q) is on the parabola, then (2·(−b/2a) − p, q) is also on it.
Practical use: once you find one root, the other root is the same distance from the axis of symmetry on the opposite side.
Real-World Applications
| Application | Equation Type | What the Vertex Gives You | |-------------|--------------|--------------------------| | Projectile motion | h = −16t² + v₀t + h₀ | Maximum height | | Profit maximization | P = −ax² + bx + c | Maximum profit | | Area optimization | A = −wx² + lx | Maximum area | | Cost minimization | C = ax² + bx + c | Minimum cost |
The Trench Truth: In physics, the vertex of a projectile's path gives you the maximum height AND the time it occurs. If h(t) = −16t² + 48t + 5, the vertex is at t = 1.5s with h = 41ft. No need to test multiple time values — the vertex formula gives you the answer directly.
Related: Derivative Calculator · Circle Calculator · Statistics Calculator
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