At Westonci.ca, we make it easy to get the answers you need from a community of informed and experienced contributors. Get detailed and accurate answers to your questions from a community of experts on our comprehensive Q&A platform. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
Sagot :
Certainly! Let's delve into the process of graphing the polynomial function [tex]\( g(x) = -x(x-3)^2(x+2)^2 \)[/tex]. We'll analyze the function step by step, focusing on its key features such as roots, end behavior, and turning points.
### Step 1: Identify the Roots
First, let's find the roots (or zeros) of the polynomial, which occur when [tex]\( g(x) = 0 \)[/tex].
Set [tex]\( g(x) = 0 \)[/tex]:
[tex]\[ -x(x-3)^2(x+2)^2 = 0 \][/tex]
This equation will be zero if any of the factors are zero:
1. [tex]\( -x = 0 \)[/tex] implies [tex]\( x = 0 \)[/tex]
2. [tex]\( (x-3)^2 = 0 \)[/tex] implies [tex]\( x = 3 \)[/tex]
3. [tex]\( (x+2)^2 = 0 \)[/tex] implies [tex]\( x = -2 \)[/tex]
Therefore, the roots are at [tex]\( x = 0 \)[/tex], [tex]\( x = 3 \)[/tex], and [tex]\( x = -2 \)[/tex].
### Step 2: Determine Multiplicity of Roots
The multiplicity of a root affects the graph's behavior at that root.
- [tex]\( x = 0 \)[/tex]: This root has multiplicity 1 (since it appears as [tex]\( x \)[/tex]).
- [tex]\( x = 3 \)[/tex]: This root has multiplicity 2 (since it appears as [tex]\( (x-3)^2 \)[/tex]).
- [tex]\( x = -2 \)[/tex]: This root has multiplicity 2 (since it appears as [tex]\( (x+2)^2 \)[/tex]).
### Step 3: End Behavior of the Polynomial
Consider the leading term to determine the polynomial's end behavior. The highest degree term in [tex]\( g(x) \)[/tex] is obtained by multiplying the leading terms in each factor:
[tex]\[ -x \cdot x^2 \cdot x^2 = -x^5 \][/tex]
As [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex]:
[tex]\[ g(x) \to -x^5 \to -\infty \][/tex]
As [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex]:
[tex]\[ g(x) \to -x^5 \to \infty \][/tex]
Therefore, the graph will fall to [tex]\( -\infty \)[/tex] as [tex]\( x \)[/tex] goes to [tex]\( \infty \)[/tex] and rise to [tex]\( \infty \)[/tex] as [tex]\( x \)[/tex] goes to [tex]\( -\infty \)[/tex].
### Step 4: Behavior Near the Roots
- At [tex]\( x = 0 \)[/tex] (multiplicity 1): The graph crosses the x-axis.
- At [tex]\( x = 3 \)[/tex] (multiplicity 2): The graph touches the x-axis and turns around, forming a local minimum or maximum.
- At [tex]\( x = -2 \)[/tex] (multiplicity 2): The graph touches the x-axis and turns around, forming a local minimum or maximum.
### Step 5: Additional Points and Sketch
To better understand the shape of the graph, evaluate the function at additional points, particularly near the roots and around where the graph turns.
#### At [tex]\( x = -3 \)[/tex]:
[tex]\[ g(-3) = -(-3) \cdot (-3-3)^2 \cdot (-3+2)^2 = 3 \cdot 36 \cdot 1 = 108 \][/tex]
#### At [tex]\( x = -1 \)[/tex]:
[tex]\[ g(-1) = -(-1) \cdot (-1-3)^2 \cdot (-1+2)^2 = 1 \cdot 16 \cdot 1 = -16 \][/tex]
#### At [tex]\( x = 1 \)[/tex]:
[tex]\[ g(1) = -1 \cdot (1-3)^2 \cdot (1+2)^2 = -1 \cdot 4 \cdot 9 = -36 \][/tex]
#### At [tex]\( x = 4 \)[/tex]:
[tex]\[ g(4) = -4 \cdot (4-3)^2 \cdot (4+2)^2 = -4 \cdot 1 \cdot 36 = -144 \][/tex]
### Sketch the Graph
With this information, we can sketch a rough graph:
1. The graph rises to [tex]\( \infty \)[/tex] as [tex]\( x \)[/tex] goes to [tex]\( -\infty \)[/tex].
2. It crosses the x-axis at [tex]\( x = 0 \)[/tex].
3. It touches and turns around at [tex]\( x = -2 \)[/tex] and [tex]\( x = 3 \)[/tex].
4. It has significant points at [tex]\( (-3, 108) \)[/tex], [tex]\( (-1, -16) \)[/tex], [tex]\( (1, -36) \)[/tex], and [tex]\( (4, -144) \)[/tex].
The overall shape should reflect these key features: starting from [tex]\( \infty \)[/tex] on the left, turning around at [tex]\( x = -2 \)[/tex], crossing at [tex]\( x = 0 \)[/tex], turning around again at [tex]\( x = 3 \)[/tex], and finally falling towards [tex]\( -\infty \)[/tex] on the right.
This completes the detailed examination of the function [tex]\( g(x) = -x(x-3)^2(x+2)^2 \)[/tex] and its graph.
### Step 1: Identify the Roots
First, let's find the roots (or zeros) of the polynomial, which occur when [tex]\( g(x) = 0 \)[/tex].
Set [tex]\( g(x) = 0 \)[/tex]:
[tex]\[ -x(x-3)^2(x+2)^2 = 0 \][/tex]
This equation will be zero if any of the factors are zero:
1. [tex]\( -x = 0 \)[/tex] implies [tex]\( x = 0 \)[/tex]
2. [tex]\( (x-3)^2 = 0 \)[/tex] implies [tex]\( x = 3 \)[/tex]
3. [tex]\( (x+2)^2 = 0 \)[/tex] implies [tex]\( x = -2 \)[/tex]
Therefore, the roots are at [tex]\( x = 0 \)[/tex], [tex]\( x = 3 \)[/tex], and [tex]\( x = -2 \)[/tex].
### Step 2: Determine Multiplicity of Roots
The multiplicity of a root affects the graph's behavior at that root.
- [tex]\( x = 0 \)[/tex]: This root has multiplicity 1 (since it appears as [tex]\( x \)[/tex]).
- [tex]\( x = 3 \)[/tex]: This root has multiplicity 2 (since it appears as [tex]\( (x-3)^2 \)[/tex]).
- [tex]\( x = -2 \)[/tex]: This root has multiplicity 2 (since it appears as [tex]\( (x+2)^2 \)[/tex]).
### Step 3: End Behavior of the Polynomial
Consider the leading term to determine the polynomial's end behavior. The highest degree term in [tex]\( g(x) \)[/tex] is obtained by multiplying the leading terms in each factor:
[tex]\[ -x \cdot x^2 \cdot x^2 = -x^5 \][/tex]
As [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex]:
[tex]\[ g(x) \to -x^5 \to -\infty \][/tex]
As [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex]:
[tex]\[ g(x) \to -x^5 \to \infty \][/tex]
Therefore, the graph will fall to [tex]\( -\infty \)[/tex] as [tex]\( x \)[/tex] goes to [tex]\( \infty \)[/tex] and rise to [tex]\( \infty \)[/tex] as [tex]\( x \)[/tex] goes to [tex]\( -\infty \)[/tex].
### Step 4: Behavior Near the Roots
- At [tex]\( x = 0 \)[/tex] (multiplicity 1): The graph crosses the x-axis.
- At [tex]\( x = 3 \)[/tex] (multiplicity 2): The graph touches the x-axis and turns around, forming a local minimum or maximum.
- At [tex]\( x = -2 \)[/tex] (multiplicity 2): The graph touches the x-axis and turns around, forming a local minimum or maximum.
### Step 5: Additional Points and Sketch
To better understand the shape of the graph, evaluate the function at additional points, particularly near the roots and around where the graph turns.
#### At [tex]\( x = -3 \)[/tex]:
[tex]\[ g(-3) = -(-3) \cdot (-3-3)^2 \cdot (-3+2)^2 = 3 \cdot 36 \cdot 1 = 108 \][/tex]
#### At [tex]\( x = -1 \)[/tex]:
[tex]\[ g(-1) = -(-1) \cdot (-1-3)^2 \cdot (-1+2)^2 = 1 \cdot 16 \cdot 1 = -16 \][/tex]
#### At [tex]\( x = 1 \)[/tex]:
[tex]\[ g(1) = -1 \cdot (1-3)^2 \cdot (1+2)^2 = -1 \cdot 4 \cdot 9 = -36 \][/tex]
#### At [tex]\( x = 4 \)[/tex]:
[tex]\[ g(4) = -4 \cdot (4-3)^2 \cdot (4+2)^2 = -4 \cdot 1 \cdot 36 = -144 \][/tex]
### Sketch the Graph
With this information, we can sketch a rough graph:
1. The graph rises to [tex]\( \infty \)[/tex] as [tex]\( x \)[/tex] goes to [tex]\( -\infty \)[/tex].
2. It crosses the x-axis at [tex]\( x = 0 \)[/tex].
3. It touches and turns around at [tex]\( x = -2 \)[/tex] and [tex]\( x = 3 \)[/tex].
4. It has significant points at [tex]\( (-3, 108) \)[/tex], [tex]\( (-1, -16) \)[/tex], [tex]\( (1, -36) \)[/tex], and [tex]\( (4, -144) \)[/tex].
The overall shape should reflect these key features: starting from [tex]\( \infty \)[/tex] on the left, turning around at [tex]\( x = -2 \)[/tex], crossing at [tex]\( x = 0 \)[/tex], turning around again at [tex]\( x = 3 \)[/tex], and finally falling towards [tex]\( -\infty \)[/tex] on the right.
This completes the detailed examination of the function [tex]\( g(x) = -x(x-3)^2(x+2)^2 \)[/tex] and its graph.
Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.
