Which of the following could be the multiplicity of a zero whose graph touches the x-axis then turns back?
1. 1
2. 2
3. 3
4. 0
The multiplicity of a zero determines the behavior of the graph near the x-axis.
1. A zero with multiplicity 1 would produce a graph that intersects the x-axis at the zero.
2. A zero with multiplicity 2 would produce a graph that touches the x-axis but does not turn back.
3. A zero with multiplicity 3 would produce a graph that touches the x-axis and turns back.
4. A zero with multiplicity 0 would not produce a graph that touches the x-axis or intersects it.
Therefore, the correct answer is 3.
The multiplicity of a zero refers to the number of times a factor is repeated in the factored form of a polynomial.
When the graph of a function touches the x-axis and then turns back, it means that the zero has an even multiplicity.
Therefore, the possible choices for the multiplicity of a zero whose graph touches the x-axis then turns back are:
1. 2 (Option 2)
To determine the multiplicity of a zero on the graph of a function, we need to analyze how the graph behaves at the x-intercept.
A zero with a multiplicity of 1 means that the graph touches the x-axis and then crosses it. This means that the graph intersects the x-axis at a single point.
A zero with a multiplicity of 2 means that the graph touches the x-axis and stays tangent to it, but does not cross it. This means that the graph intersects the x-axis and then turns back, remaining in contact with the x-axis at that point.
A zero with a multiplicity of 3 means that the graph touches the x-axis and remains tangent to it with no change in direction. This means that the graph intersects the x-axis and stays in contact without turning back.
A zero with a multiplicity of 0 means that the graph does not touch the x-axis and does not intersect it.
Given the behavior described in the question - touching the x-axis and then turning back - the correct choice would be 2, which represents a zero with a multiplicity of 2.