Identify which graphs are not those of polynomial functions.

Verified step by step guidance

1. Step 1: Observe the shape of the graph. The graph shows a V-shaped function with two straight lines meeting at the origin (0,0).
2. Step 2: Recall that polynomial functions are smooth and continuous with no sharp corners or cusps. Their graphs are made up of curves, not sharp angles.
3. Step 3: Notice that the graph has a sharp corner at the origin, which indicates a cusp or a point where the function is not differentiable.
4. Step 4: Recognize that this V-shaped graph resembles the absolute value function, which is defined as \(y = |x|\), and is not a polynomial because it involves an absolute value operation.
5. Step 5: Conclude that since the graph has a sharp corner and is not smooth, it is not the graph of a polynomial 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, and they can have turning points but no abrupt changes in direction.

Graph Characteristics of Polynomial Functions

The graphs of polynomial functions are continuous and differentiable everywhere, meaning they have no sharp points or corners. They typically exhibit smooth curves, and their end behavior depends on the leading term's degree and coefficient.

Non-Polynomial Graphs and Sharp Corners

Graphs with sharp corners or 'V' shapes, like the absolute value function, are not polynomial functions because polynomials are differentiable everywhere. Such graphs indicate piecewise or non-polynomial functions, which have points where the derivative does not exist.