Use the image to answer the question.

Question

A plotted curve with 3 turning points is labeled w left parenthesis x right parenthesis. The plot starts in quadrant 3, rises to a turning point on the x-axis and slowly rises to a turning point in quadrant 1. It descends to the x-axis and then rises steeply.

(2x+1)
is a factor of w(x), whose graph is shown. Is the multiplicity of (2x+1)
even or odd?

Option #1: even

Option #2: odd

(1 point)
The multiplicity of (2x+1)
is described by Option #
.

Answer 1

To determine the multiplicity of the factor (2x + 1) based on the behavior of the graph at the x-axis, we can analyze how the graph interacts with the x-axis at its corresponding root.

1. Identify the root: The factor (2x + 1) = 0 gives us the root x = -1/2.

2. Observe the graph: At x = -1/2, if the graph touches the x-axis and turns back without crossing it (i.e., it bounces off the axis), then the multiplicity is even. If the graph crosses the x-axis at that point, then the multiplicity is odd.

Since you mentioned that the plot shows the curve starting in quadrant 3, rising to the x-axis (where it presumably has a turning point), descending to the x-axis again, and rising steeply, it does not indicate that it bounces at this root. Instead, it suggests that there is a crossing at the x-axis.

Therefore, the multiplicity of (2x + 1) is odd.

Option #2: odd.