Inputs
Results
Slope m
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Point used
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f(x) at point
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Vertex (xᵥ, yᵥ)
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Tangent line
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Derivative y′(x) = 2ax + b
Average slope = [f(x₂) − f(x₁)] / (x₂ − x₁)
Tip: Use “Add to Results” to build a dataset.
Results Table
| # | Mode | a | b | c | x / x₁ | x₂ / m* | m | f(x) | Tangent line | Vertex xᵥ | Vertex yᵥ | Units | Time |
|---|
Example Data
| a | b | c | Mode | x / x₁ | x₂ / m* | Action |
|---|---|---|---|---|---|---|
| 1 | -3 | 2 | instant | 1.5 | — | |
| 2 | 0 | -1 | average | 0 | 2 | |
| -0.5 | 4 | 0 | vertex | — | — | |
| 0.75 | -1 | 3 | target | — | 0 |
Formulas Used
- Quadratic: y(x) = a x² + b x + c
- Derivative (slope at x): y′(x) = 2 a x + b
- Average slope on [x₁, x₂]: m = [f(x₂) − f(x₁)] / (x₂ − x₁)
- Vertex: xᵥ = −b/(2a), yᵥ = f(xᵥ)
- Target slope m*: solve 2 a x + b = m* → x = (m* − b)/(2a)
- Tangent line at (x₀, f(x₀)): y = m (x − x₀) + f(x₀)
Notes. Requires a ≠ 0. For average slope, ensure x₂ ≠ x₁. Units are carried symbolically as [y-unit per x-unit].
How to Use
- Enter coefficients a, b, c for your quadratic.
- Select a mode and fill only the relevant fields.
- Optionally set unit labels and decimal places.
- Click Calculate to preview results.
- Click Add to Results to append the row to the table.
- Use Export CSV or Download PDF to save.
Pro tip: The vertex is where the instantaneous slope equals zero.
Worked Example
Function: y(x) = x² − 3x + 2 (a = 1, b = −3, c = 2)
- Instantaneous slope at x = 1.5.
Derivative y′(x) = 2x − 3 ⇒ y′(1.5) = 0. - Evaluate point: f(1.5) = 1.5² − 3·1.5 + 2 = −0.25.
- Tangent line: y = 0·(x − 1.5) + (−0.25) = −0.25.
- Vertex: xᵥ = −b/(2a) = 1.5, yᵥ = −0.25 (slope zero at vertex).
Average slope on [0, 2]: f(0)=2, f(2)=0 ⇒ m = (0 − 2)/(2 − 0) = −1.
Target slope m* = 2: Solve 2a x + b = m* ⇒ 2x − 3 = 2 ⇒ x = 2.5.