How does multiplicity affect the shape of a graph at an x-intercept?

Multiplicity refers to the number of times a particular root occurs in a polynomial. When examining how multiplicity affects the shape of a graph at an x-intercept, it is crucial to consider the behavior of the graph as it approaches the x-axis at that point.

When the multiplicity of a root is 1 (an odd multiplicity), the graph crosses the x-axis at that intercept. This is because as the polynomial approaches that intercept, it does so from one side of the axis and leaves on the opposite side. This crossing behavior is evident because the polynomial changes sign at that point.

For multiplicities greater than 1 (even multiplicities), the behavior is different. An even multiplicity indicates that the graph will touch the x-axis at the intercept but will not cross it. Instead, it will either rise or fall before and after reaching the intercept, resulting in the graph appearing to bounce off the axis.

Thus, the effect of multiplicity specifically indicates that a multiplicity of 1 leads to a crossing of the graph at the x-intercept, which is why this choice is the correct one. This understanding helps in predicting the shape of the polynomial graph and can aid in sketching or analyzing polynomial functions effectively.