Smooth curve

Definition

A smooth curve is a continuous curve that has no breaks, sharp corners, or cusps. In algebra and trigonometry, it typically refers to the graph of a polynomial function which is differentiable at all points.

5 Must Know Facts For Your Next Test

1. A smooth curve representing a polynomial function of degree $n$ will have at most $n-1$ turning points.
2. The smoothness of a curve ensures that its first derivative exists and is continuous.
3. Inflection points on a smooth curve occur where the second derivative changes sign.
4. Polynomial functions form smooth curves because they are continuous and differentiable everywhere on their domain.
5. Higher-degree polynomials can produce more complex smooth curves with multiple peaks and valleys.

Review Questions

• What characteristics make a curve 'smooth'?
• How many turning points can you expect in the graph of a third-degree polynomial?
• Why are polynomial functions always represented by smooth curves?

Related terms

Polynomial Function: A mathematical expression involving a sum of powers in one or more variables multiplied by coefficients.

Continuous: A function is continuous if there are no breaks, holes, or gaps in its graph.

Differentiable: A function is differentiable if it has a derivative at every point in its domain.