To describe the behavior of the graph of the function [tex]\( f(x) = (x-2)^3(x+6)^2(x+12) \)[/tex] at its roots, let's analyze each root in turn.
1. At [tex]\( x = 2 \)[/tex]:
- The term [tex]\((x-2)^3\)[/tex] contributes to the function becoming zero at [tex]\( x = 2 \)[/tex].
- The exponent 3 is an odd number, which means that at this root, the graph will touch the x-axis and change direction (cross the x-axis).
Therefore, at [tex]\( x=2 \)[/tex], the graph touches and crosses the x-axis.
2. At [tex]\( x = -6 \)[/tex]:
- The term [tex]\((x+6)^2\)[/tex] results in the function becoming zero at [tex]\( x = -6 \)[/tex].
- The exponent 2 is an even number, implying that at this root, the graph will touch the x-axis but will not change direction (it does not cross the x-axis).
Hence, at [tex]\( x=-6 \)[/tex], the graph touches but does not cross the x-axis.
3. At [tex]\( x = -12 \)[/tex]:
- The term [tex]\((x+12)\)[/tex] causes the function to be zero at [tex]\( x = -12 \)[/tex].
- The exponent of this term is 1, which is an odd number. This means that at this root, the graph will touch the x-axis and change direction (cross the x-axis).
Consequently, at [tex]\( x=-12 \)[/tex], the graph touches and crosses the x-axis.
Summarizing our findings:
- At [tex]\( x=2 \)[/tex], the graph touches and crosses the x-axis.
- At [tex]\( x=-6 \)[/tex], the graph touches but does not cross the x-axis.
- At [tex]\( x=-12 \)[/tex], the graph touches and crosses the x-axis.
So the completed statement is:
At [tex]\( x=2 \)[/tex], the graph touches and crosses the [tex]\( x \)[/tex]-axis.
At [tex]\( x=-6 \)[/tex], the graph touches but does not cross the [tex]\( x \)[/tex]-axis.
At [tex]\( x=-12 \)[/tex], the graph touches and crosses the [tex]\( x \)[/tex]-axis.
