Question
Question content area top left
Part 1
Answer parts (a)dash(e) for the function shown below.
f(x)equals x cubed plus 3 x squared minus x minus 3
Question content area bottom left
Part 1
a. Use the leading coefficient test to determine the graph's end behavior. Which statement describes the behavior at the ends of f(x)equals x cubed plus 3 x squared minus x minus 3?
A.
The graph falls to the left and rises to the right.
B.
The graph rises to the left and to the right.
C.
The graph rises to the left and falls to the right.
D.
The graph falls to the left and to the right.
.
.
.
Question content area right
Part 1
-10
-8
-6
-4
-2
2
4
6
8
10
-10
-8
-6
-4
-2
2
4
6
8
10
x
y
b. Find the x-intercepts. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept. What are the x-intercepts?
xequals
enter your response here
(Type an integer or a decimal. Use a comma to separate answers as needed.)
At which x-intercept(s) does the graph cross the x-axis? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A.
xequals
enter your response here
(Type an integer or a decimal. Use a comma to separate answers as needed.)
B.
There are no x-intercepts at which the graph crosses the x-axis.
Part 1
Answer parts (a)dash(e) for the function shown below.
f(x)equals x cubed plus 3 x squared minus x minus 3
Question content area bottom left
Part 1
a. Use the leading coefficient test to determine the graph's end behavior. Which statement describes the behavior at the ends of f(x)equals x cubed plus 3 x squared minus x minus 3?
A.
The graph falls to the left and rises to the right.
B.
The graph rises to the left and to the right.
C.
The graph rises to the left and falls to the right.
D.
The graph falls to the left and to the right.
.
.
.
Question content area right
Part 1
-10
-8
-6
-4
-2
2
4
6
8
10
-10
-8
-6
-4
-2
2
4
6
8
10
x
y
b. Find the x-intercepts. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept. What are the x-intercepts?
xequals
enter your response here
(Type an integer or a decimal. Use a comma to separate answers as needed.)
At which x-intercept(s) does the graph cross the x-axis? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A.
xequals
enter your response here
(Type an integer or a decimal. Use a comma to separate answers as needed.)
B.
There are no x-intercepts at which the graph crosses the x-axis.
Answers
GPT-4o mini
To analyze the function \( f(x) = x^3 + 3x^2 - x - 3 \) and answer the questions, let's go through parts (a) and (b) step by step.
### Part (a): End Behavior Using Leading Coefficient Test
1. **Identify the Degree and Leading Coefficient:**
- The degree of the polynomial is 3 (since the highest power of \( x \) is 3).
- The leading coefficient is 1 (the coefficient of \( x^3 \)).
2. **Determine the End Behavior:**
- For a polynomial function:
- If the degree is odd and the leading coefficient is positive, the graph falls to the left and rises to the right.
- If the degree is odd and the leading coefficient is negative, the graph rises to the left and falls to the right.
Given our leading coefficient is positive and the degree is odd:
- **Conclusion**: The graph falls to the left and rises to the right.
**Answer for Part (a)**:
A. The graph falls to the left and rises to the right.
---
### Part (b): Finding the x-Intercepts
To find the x-intercepts, we need to solve for \( x \) when \( f(x) = 0 \):
1. **Set the equation to zero:**
\[
x^3 + 3x^2 - x - 3 = 0
\]
2. **Finding x-intercepts (Factoring or using the Rational Root Theorem):**
- We can try to find rational roots using trial and error with factors of -3 (the constant term).
- Testing \( x = 1 \):
\[
f(1) = 1^3 + 3(1^2) - 1 - 3 = 1 + 3 - 1 - 3 = 0
\]
So, \( x = 1 \) is a root.
3. **Divide \( f(x) \) by \( (x - 1) \):**
We can use synthetic division or polynomial long division to find the other factors:
\[
f(x) = (x - 1)(x^2 + 4x + 3)
\]
4. **Factor the quadratic \( x^2 + 4x + 3 \):**
\[
x^2 + 4x + 3 = (x + 1)(x + 3)
\]
5. **Write the fully factored form:**
\[
f(x) = (x - 1)(x + 1)(x + 3)
\]
6. **Setting each factor to zero for x-intercepts:**
- \( x - 1 = 0 \) gives \( x = 1 \)
- \( x + 1 = 0 \) gives \( x = -1 \)
- \( x + 3 = 0 \) gives \( x = -3 \)
Thus, the x-intercepts are \( x = 1, -1, -3 \).
7. **Determining if the graph crosses or touches the x-axis:**
- The roots \( x = 1 \), \( x = -1 \), and \( x = -3 \) are all simple roots since they are single factors. Therefore, the graph crosses the x-axis at each of these intercepts.
**Answer for Part (b)**:
- The x-intercepts are: \( x = 1, -1, -3 \).
- At which \( x \)-intercept(s) does the graph cross the x-axis?
- A. \( x = 1, -1, -3 \).
### Final Answers:
- (a) A. The graph falls to the left and rises to the right.
- (b) x-intercepts: \( 1, -1, -3 \) and it crosses at \( 1, -1, -3 \).
### Part (a): End Behavior Using Leading Coefficient Test
1. **Identify the Degree and Leading Coefficient:**
- The degree of the polynomial is 3 (since the highest power of \( x \) is 3).
- The leading coefficient is 1 (the coefficient of \( x^3 \)).
2. **Determine the End Behavior:**
- For a polynomial function:
- If the degree is odd and the leading coefficient is positive, the graph falls to the left and rises to the right.
- If the degree is odd and the leading coefficient is negative, the graph rises to the left and falls to the right.
Given our leading coefficient is positive and the degree is odd:
- **Conclusion**: The graph falls to the left and rises to the right.
**Answer for Part (a)**:
A. The graph falls to the left and rises to the right.
---
### Part (b): Finding the x-Intercepts
To find the x-intercepts, we need to solve for \( x \) when \( f(x) = 0 \):
1. **Set the equation to zero:**
\[
x^3 + 3x^2 - x - 3 = 0
\]
2. **Finding x-intercepts (Factoring or using the Rational Root Theorem):**
- We can try to find rational roots using trial and error with factors of -3 (the constant term).
- Testing \( x = 1 \):
\[
f(1) = 1^3 + 3(1^2) - 1 - 3 = 1 + 3 - 1 - 3 = 0
\]
So, \( x = 1 \) is a root.
3. **Divide \( f(x) \) by \( (x - 1) \):**
We can use synthetic division or polynomial long division to find the other factors:
\[
f(x) = (x - 1)(x^2 + 4x + 3)
\]
4. **Factor the quadratic \( x^2 + 4x + 3 \):**
\[
x^2 + 4x + 3 = (x + 1)(x + 3)
\]
5. **Write the fully factored form:**
\[
f(x) = (x - 1)(x + 1)(x + 3)
\]
6. **Setting each factor to zero for x-intercepts:**
- \( x - 1 = 0 \) gives \( x = 1 \)
- \( x + 1 = 0 \) gives \( x = -1 \)
- \( x + 3 = 0 \) gives \( x = -3 \)
Thus, the x-intercepts are \( x = 1, -1, -3 \).
7. **Determining if the graph crosses or touches the x-axis:**
- The roots \( x = 1 \), \( x = -1 \), and \( x = -3 \) are all simple roots since they are single factors. Therefore, the graph crosses the x-axis at each of these intercepts.
**Answer for Part (b)**:
- The x-intercepts are: \( x = 1, -1, -3 \).
- At which \( x \)-intercept(s) does the graph cross the x-axis?
- A. \( x = 1, -1, -3 \).
### Final Answers:
- (a) A. The graph falls to the left and rises to the right.
- (b) x-intercepts: \( 1, -1, -3 \) and it crosses at \( 1, -1, -3 \).