To determine if a polynomial graph bounces, you need to analyze the behavior of the graph at its roots or x-intercepts. The key concept here is the multiplicity of the roots.
When a polynomial has a root (or zero) at a certain value of x, the behavior of the graph at that point depends on the multiplicity of the root:
- Odd Multiplicity: If a polynomial has a root with an odd multiplicity (like 1, 3, 5, etc.), the graph will cross the x-axis at that root, meaning it does not bounce. Instead, it smoothly transitions through the x-axis.
- Even Multiplicity: If a polynomial has a root with an even multiplicity (like 2, 4, 6, etc.), the graph will bounce off the x-axis at that root. This means that as the graph approaches the root from one side, it will touch the x-axis and then turn back in the opposite direction.
To summarize, check the multiplicity of the roots of the polynomial:
- Factor the polynomial to find its roots.
- Determine the multiplicity of each root.
- If the multiplicity is even, expect the graph to bounce; if it is odd, expect the graph to cross the axis.
This analysis will help you visually understand the behavior of polynomial graphs at their x-intercepts.