To understand the behavior of the graph of the function [tex]\( g(x) = (x + 4)^4 (x - 9) \)[/tex] at the points [tex]\( x = -4 \)[/tex] and [tex]\( x = 9 \)[/tex], we need to analyze the given function's factors and their exponents.
1. Analyzing behavior at [tex]\( x = -4 \)[/tex]:
- The factor that is significant at [tex]\( x = -4 \)[/tex] is [tex]\( (x + 4)^4 \)[/tex].
- When [tex]\( x = -4 \)[/tex], the term [tex]\( (x + 4) \)[/tex] becomes zero, so [tex]\( (x + 4)^4 \)[/tex] also becomes zero.
- The exponent of this factor is 4, which is an even exponent.
- For factors with even exponents, the graph touches the x-axis at the corresponding root but does not cross it. This is because raising a number to an even power ensures the product is non-negative, and the sign of the expression does not change as it passes through that root.
Therefore, at [tex]\( x = -4 \)[/tex], the graph touches the x-axis.
2. Analyzing behavior at [tex]\( x = 9 \)[/tex]:
- The factor that is significant at [tex]\( x = 9 \)[/tex] is [tex]\( (x - 9) \)[/tex].
- When [tex]\( x = 9 \)[/tex], the term [tex]\( (x - 9) \)[/tex] becomes zero, so [tex]\( (x - 9) \)[/tex] also becomes zero.
- The exponent of this factor is 1, which is an odd exponent.
- For factors with odd exponents, the graph crosses the x-axis at the corresponding root. This is because raising a number to an odd power preserves the sign of the product, causing the graph to change its sign and, thus, pass through the x-axis.
Therefore, at [tex]\( x = 9 \)[/tex], the graph crosses the x-axis.
In conclusion:
- At [tex]\( x = -4 \)[/tex], the graph touches the x-axis.
- At [tex]\( x = 9 \)[/tex], the graph crosses the x-axis.
