Parabola and the tangent line at a moving point
MathematicsCalculusAges 16–17
Use it with your class: students open it from a link or a QR code, with no account needed.
Next in Calculus
The graph of y = ax² + bx + c has a point M that moves along the parabola. Drag M with the mouse or a finger; the tangent line at M is drawn from the slope y'(x) = 2ax + b, showing the coordinates of M, the slope, and the equation of the tangent line.
Lesson: Geometric meaning of the derivative, tangent lines
What it shows
The derivative at x₀ is the slope of the tangent to the graph at the point with x-coordinate x₀. For the parabola y = ax² + bx + c the derivative y′ = 2ax + b is linear, so the slope changes at a constant rate as the point moves. The tangent at M(x₀; y₀) is y = y′(x₀)(x − x₀) + y₀. It is horizontal exactly at the vertex, where 2ax₀ + b = 0, and it stays on one side of the parabola, touching it only at M: a better picture of tangency than meeting the curve once.
How to use
Set Coefficient a, Coefficient b and Coefficient c to choose the parabola and Initial x-coordinate of M to place the point. Students then drag M along the curve and track the slope k and the tangent line equation shown below the graph. Ask them to find where the tangent is horizontal, where the slope is negative, and how k changes as M moves from left to right.
Parameters you can change
- Coefficient a -5–5
- Coefficient b -10–10
- Coefficient c -10–10
- Initial x-coordinate of M -8–8
Questions to explore
- At which point of the parabola is the tangent parallel to the x-axis, and how is that point related to the vertex?
- What is the equation of the tangent to y = x² − 2x − 3 at x₀ = 2?
- How does the slope k change as M moves from left to right, and what changes when a is negative?
