Let's analyze the roots of the function [tex]\( f(x) = (x-2)^3(x+6)^2(x+12) \)[/tex] and describe the behavior of the graph at these roots.
To understand how the graph behaves at each root, we need to examine the exponents of the factors corresponding to each root. Here are the steps to determine the behavior:
1. Identify the roots and their exponents:
- The factor [tex]\((x-2)^3\)[/tex] has a root at [tex]\( x = 2 \)[/tex] with an exponent of 3.
- The factor [tex]\((x+6)^2\)[/tex] has a root at [tex]\( x = -6 \)[/tex] with an exponent of 2.
- The factor [tex]\((x+12)\)[/tex] has a root at [tex]\( x = -12 \)[/tex] with an exponent of 1.
2. Determine the behavior at each root:
- For [tex]\( x = 2 \)[/tex]:
- The exponent of the factor is 3, which is odd. When the exponent is odd, the graph crosses the x-axis at that root.
- Therefore, at [tex]\( x = 2 \)[/tex], the graph crosses the x-axis.
- For [tex]\( x = -6 \)[/tex]:
- The exponent of the factor is 2, which is even. When the exponent is even, the graph touches the x-axis and turns around at that root.
- Therefore, at [tex]\( x = -6 \)[/tex], the graph touches the x-axis.
- For [tex]\( x = -12 \)[/tex]:
- The exponent of the factor is 1, which is odd. When the exponent is odd, the graph crosses the x-axis at that root.
- Therefore, at [tex]\( x = -12 \)[/tex], the graph crosses the x-axis.
Summarizing the behavior of the graph at each root:
- At [tex]\( x = 2 \)[/tex], the graph crosses the x-axis.
- At [tex]\( x = -6 \)[/tex], the graph touches the x-axis.
- At [tex]\( x = -12 \)[/tex], the graph crosses the x-axis.
