To determine the end behavior of the function [tex]\( f(x) = x^5 - 8x^4 + 16x^3 \)[/tex], we need to analyze the term with the highest power, as it dominates the behavior of the function for very large or very small values of [tex]\( x \)[/tex].
### Step 1: End Behavior Analysis
The highest power term in [tex]\( f(x) \)[/tex] is [tex]\( x^5 \)[/tex]:
- As [tex]\( x \to -\infty \)[/tex]:
- The term [tex]\( x^5 \)[/tex] will approach [tex]\( -\infty \)[/tex].
- Therefore, [tex]\( f(x) \to -\infty \)[/tex].
- As [tex]\( x \to +\infty \)[/tex]:
- The term [tex]\( x^5 \)[/tex] will approach [tex]\( +\infty \)[/tex].
- Therefore, [tex]\( f(x) \to +\infty \)[/tex].
Based on this analysis, the end behavior of the function is:
[tex]\[ f(x) \rightarrow -\infty \text{ as } x \rightarrow -\infty \][/tex]
[tex]\[ f(x) \rightarrow +\infty \text{ as } x \rightarrow +\infty \][/tex]
### Step 2: Identifying Roots
To find the points where the graph touches or crosses the x-axis, we need to solve for [tex]\( f(x) = 0 \)[/tex]:
[tex]\[ x^5 - 8x^4 + 16x^3 = 0 \][/tex]
Factoring out the common term [tex]\( x^3 \)[/tex]:
[tex]\[ x^3(x^2 - 8x + 16) = 0 \][/tex]
Solving the factored equation:
1. [tex]\( x^3 = 0 \)[/tex] gives roots [tex]\( x = 0 \)[/tex] (with multiplicity 3).
2. [tex]\( x^2 - 8x + 16 = 0 \)[/tex] is a quadratic equation. Factoring, we get:
[tex]\[ (x - 4)^2 = 0 \][/tex]
This gives a root [tex]\( x = 4 \)[/tex] (with multiplicity 2).
### Step 3: Behavior at the Roots
- The root [tex]\( x = 0 \)[/tex] has a multiplicity of 3, which means the function will touch the x-axis at this point without crossing it.
- The root [tex]\( x = 4 \)[/tex] has a multiplicity of 2, indicating the function will also touch the x-axis at this point without crossing it.
### Summary
The correct end behavior and the behavior at the roots can be stated as follows:
- [tex]\( f(x) \to -\infty \)[/tex] as [tex]\( x \to -\infty \)[/tex]
- [tex]\( f(x) \to +\infty \)[/tex] as [tex]\( x \to +\infty \)[/tex]
- The graph touches, but does not cross, the x-axis at [tex]\( x = 4 \)[/tex]
- The graph crosses the x-axis at [tex]\( x = 0 \)[/tex]
Thus, filling in the initial problem's statements:
- End behavior:
[tex]\[
f(x) \rightarrow -\infty \text{ as } x \rightarrow -\infty ; \quad f(x) \rightarrow +\infty \text{ as } x \rightarrow +\infty
\][/tex]
- The graph touches, but does not cross, the x-axis at [tex]\( x = 4 \)[/tex].
- The graph crosses the x-axis at [tex]\( x = 0 \)[/tex].
