Answer :
Let's analyze the polynomial function [tex]\( f(x) = (x + 1)^2 (x - 2) \)[/tex] and answer the questions step by step.
### (a) Choose the End Behavior of the Graph of [tex]\( f \)[/tex]:
To determine the end behavior of the polynomial, we need to analyze the highest degree term of the polynomial.
1. Highest Degree Term:
The polynomial [tex]\( f(x) = (x + 1)^2 (x - 2) \)[/tex] when expanded is a cubic polynomial. The highest degree term is determined by multiplying the highest degree terms from each factor:
[tex]\[ (x + 1)^2 (x - 2) = (x^2 + 2x + 1)(x - 2) \approx x^3 \quad \text{(when considering the dominant term)} \][/tex]
2. Leading Coefficient:
The leading term in the expanded polynomial [tex]\( x^3 + \text{(lower degree terms)} \)[/tex] is [tex]\( x^3 \)[/tex]. The coefficient of [tex]\( x^3 \)[/tex] is positive (1).
End Behavior Conclusion:
- As [tex]\( x \)[/tex] approaches positive infinity ([tex]\( x \to \infty \)[/tex]), [tex]\( f(x) \to \infty \)[/tex].
- As [tex]\( x \)[/tex] approaches negative infinity ([tex]\( x \to -\infty \)[/tex]), [tex]\( f(x) \to -\infty \)[/tex].
So, the correct end behavior of [tex]\( f \)[/tex] is:
- For large positive [tex]\( x \)[/tex], [tex]\( f(x) \to \infty \)[/tex].
- For large negative [tex]\( x \)[/tex], [tex]\( f(x) \to -\infty \)[/tex].
### (b) List Each Real Zero of [tex]\( f \)[/tex] According to the Behavior of the Graph at the X-axis Near that Zero:
To find the real zeros (roots) of the polynomial, we set [tex]\( f(x) = 0 \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[ (x + 1)^2 (x - 2) = 0 \][/tex]
This equation gives us:
- [tex]\( x = -1 \)[/tex] with multiplicity 2
- [tex]\( x = 2 \)[/tex] with multiplicity 1
Behavior of the Graph:
- Zero where the graph crosses the x-axis:
The graph crosses the x-axis at [tex]\( x = 2 \)[/tex] because the multiplicity here is 1, and this means the graph changes sign as it passes through this point.
- Zero where the graph touches, but does not cross, the x-axis:
The graph touches but does not cross the x-axis at [tex]\( x = -1 \)[/tex] because the multiplicity here is 2, indicating that the graph just touches the x-axis and turns around without crossing it.
### (c) Find the Y-Intercept of the Graph of [tex]\( f \)[/tex]:
The y-intercept occurs when [tex]\( x = 0 \)[/tex]. We substitute [tex]\( x = 0 \)[/tex] into the polynomial:
[tex]\[ f(0) = (0 + 1)^2 (0 - 2) = 1^2 \cdot (-2) = 1 \cdot (-2) = -2 \][/tex]
So, the y-intercept of [tex]\( f \)[/tex] is [tex]\( (0, -2) \)[/tex].
### (d) Graph [tex]\( f(x) = - (x + 1)^2 (x - 2) \)[/tex]:
Now we consider the function [tex]\( g(x) = - (x + 1)^2 (x - 2) \)[/tex].
1. Zeros of [tex]\( g \)[/tex]:
The zeros are the same as those of [tex]\( f \)[/tex]: [tex]\( x = -1 \)[/tex] and [tex]\( x = 2 \)[/tex].
2. End Behavior:
The leading term is now [tex]\(-x^3\)[/tex], so the end behavior is:
- As [tex]\( x \to \infty \)[/tex], [tex]\( g(x) \to -\infty \)[/tex].
- As [tex]\( x \to -\infty \)[/tex], [tex]\( g(x) \to \infty \)[/tex].
3. Behavior at Zeros:
- Zero where the graph crosses the x-axis:
The graph crosses at [tex]\( x = 2 \)[/tex].
- Zero where the graph touches, but does not cross, the x-axis:
The graph touches but does not cross at [tex]\( x = -1 \)[/tex].
4. Y-Intercept:
For [tex]\( g(x) \)[/tex], the y-intercept is:
[tex]\[ g(0) = - (0 + 1)^2 (0 - 2) = - 1^2 (-2) = - 1 \cdot (-2) = 2 \][/tex]
Graphing Instructions:
- Plot the points of intersection with the x-axis: [tex]\((-1, 0)\)[/tex] and [tex]\( (2, 0) \)[/tex].
- Plot the point of intersection with the y-axis: [tex]\( (0, 2) \)[/tex].
- For the point [tex]\( x = 2 \)[/tex], the graph crosses the x-axis.
- For the point [tex]\( x = -1 \)[/tex], the graph touches but does not cross the x-axis.
- Draw the graph considering these behaviors and the end behavior: as [tex]\( x \to \infty \)[/tex], [tex]\( g(x) \to -\infty \)[/tex]; as [tex]\( x \to -\infty \)[/tex], [tex]\( g(x) \to \infty \)[/tex].
These steps provide a detailed understanding of the polynomial function and ensure you can graph it accurately.
### (a) Choose the End Behavior of the Graph of [tex]\( f \)[/tex]:
To determine the end behavior of the polynomial, we need to analyze the highest degree term of the polynomial.
1. Highest Degree Term:
The polynomial [tex]\( f(x) = (x + 1)^2 (x - 2) \)[/tex] when expanded is a cubic polynomial. The highest degree term is determined by multiplying the highest degree terms from each factor:
[tex]\[ (x + 1)^2 (x - 2) = (x^2 + 2x + 1)(x - 2) \approx x^3 \quad \text{(when considering the dominant term)} \][/tex]
2. Leading Coefficient:
The leading term in the expanded polynomial [tex]\( x^3 + \text{(lower degree terms)} \)[/tex] is [tex]\( x^3 \)[/tex]. The coefficient of [tex]\( x^3 \)[/tex] is positive (1).
End Behavior Conclusion:
- As [tex]\( x \)[/tex] approaches positive infinity ([tex]\( x \to \infty \)[/tex]), [tex]\( f(x) \to \infty \)[/tex].
- As [tex]\( x \)[/tex] approaches negative infinity ([tex]\( x \to -\infty \)[/tex]), [tex]\( f(x) \to -\infty \)[/tex].
So, the correct end behavior of [tex]\( f \)[/tex] is:
- For large positive [tex]\( x \)[/tex], [tex]\( f(x) \to \infty \)[/tex].
- For large negative [tex]\( x \)[/tex], [tex]\( f(x) \to -\infty \)[/tex].
### (b) List Each Real Zero of [tex]\( f \)[/tex] According to the Behavior of the Graph at the X-axis Near that Zero:
To find the real zeros (roots) of the polynomial, we set [tex]\( f(x) = 0 \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[ (x + 1)^2 (x - 2) = 0 \][/tex]
This equation gives us:
- [tex]\( x = -1 \)[/tex] with multiplicity 2
- [tex]\( x = 2 \)[/tex] with multiplicity 1
Behavior of the Graph:
- Zero where the graph crosses the x-axis:
The graph crosses the x-axis at [tex]\( x = 2 \)[/tex] because the multiplicity here is 1, and this means the graph changes sign as it passes through this point.
- Zero where the graph touches, but does not cross, the x-axis:
The graph touches but does not cross the x-axis at [tex]\( x = -1 \)[/tex] because the multiplicity here is 2, indicating that the graph just touches the x-axis and turns around without crossing it.
### (c) Find the Y-Intercept of the Graph of [tex]\( f \)[/tex]:
The y-intercept occurs when [tex]\( x = 0 \)[/tex]. We substitute [tex]\( x = 0 \)[/tex] into the polynomial:
[tex]\[ f(0) = (0 + 1)^2 (0 - 2) = 1^2 \cdot (-2) = 1 \cdot (-2) = -2 \][/tex]
So, the y-intercept of [tex]\( f \)[/tex] is [tex]\( (0, -2) \)[/tex].
### (d) Graph [tex]\( f(x) = - (x + 1)^2 (x - 2) \)[/tex]:
Now we consider the function [tex]\( g(x) = - (x + 1)^2 (x - 2) \)[/tex].
1. Zeros of [tex]\( g \)[/tex]:
The zeros are the same as those of [tex]\( f \)[/tex]: [tex]\( x = -1 \)[/tex] and [tex]\( x = 2 \)[/tex].
2. End Behavior:
The leading term is now [tex]\(-x^3\)[/tex], so the end behavior is:
- As [tex]\( x \to \infty \)[/tex], [tex]\( g(x) \to -\infty \)[/tex].
- As [tex]\( x \to -\infty \)[/tex], [tex]\( g(x) \to \infty \)[/tex].
3. Behavior at Zeros:
- Zero where the graph crosses the x-axis:
The graph crosses at [tex]\( x = 2 \)[/tex].
- Zero where the graph touches, but does not cross, the x-axis:
The graph touches but does not cross at [tex]\( x = -1 \)[/tex].
4. Y-Intercept:
For [tex]\( g(x) \)[/tex], the y-intercept is:
[tex]\[ g(0) = - (0 + 1)^2 (0 - 2) = - 1^2 (-2) = - 1 \cdot (-2) = 2 \][/tex]
Graphing Instructions:
- Plot the points of intersection with the x-axis: [tex]\((-1, 0)\)[/tex] and [tex]\( (2, 0) \)[/tex].
- Plot the point of intersection with the y-axis: [tex]\( (0, 2) \)[/tex].
- For the point [tex]\( x = 2 \)[/tex], the graph crosses the x-axis.
- For the point [tex]\( x = -1 \)[/tex], the graph touches but does not cross the x-axis.
- Draw the graph considering these behaviors and the end behavior: as [tex]\( x \to \infty \)[/tex], [tex]\( g(x) \to -\infty \)[/tex]; as [tex]\( x \to -\infty \)[/tex], [tex]\( g(x) \to \infty \)[/tex].
These steps provide a detailed understanding of the polynomial function and ensure you can graph it accurately.
