In Exercises 11–14, identify which graphs are not those of polynomial functions.

Verified step by step guidance

1. Step 1: Understand the characteristics of polynomial functions. Polynomial functions are smooth and continuous, with graphs that have no sharp corners or cusps.
2. Step 2: Observe the given graph carefully. Notice that the graph has a sharp 'V' shape at the peak, which indicates a corner point.
3. Step 3: Recall that polynomial functions cannot have sharp corners or cusps because their derivatives are continuous everywhere.
4. Step 4: Since the graph has a sharp corner, it cannot be the graph of a polynomial function.
5. Step 5: Conclude that this graph represents a function that is not polynomial, such as an absolute value function or a piecewise linear function.

Key Concepts

Polynomial Functions

Polynomial functions are algebraic expressions consisting of variables raised to non-negative integer powers with constant coefficients. Their graphs are smooth and continuous curves without sharp corners or cusps. Examples include linear, quadratic, and cubic functions.

Graph Characteristics of Polynomial Functions

The graphs of polynomial functions are continuous and differentiable everywhere, meaning they have no breaks, holes, or sharp points. They exhibit smooth curves and can change direction but do not have sharp angles or 'V' shapes.

Identifying Non-Polynomial Graphs

Graphs that show sharp corners, cusps, or discontinuities are not polynomial functions. For example, a 'V'-shaped graph, like the absolute value function, is not polynomial because it is not differentiable at the vertex, indicating a sharp corner.