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
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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.
At which x-intercept(s) does the graph touch the x-axis and turn around? 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 touches the x-axis and turns around.
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.
At which x-intercept(s) does the graph touch the x-axis and turn around? 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 touches the x-axis and turns around.
Answers
GPT-4o mini
To analyze the function \( f(x) = x^3 + 3x^2 - x - 3 \), we will answer the questions step by step.
### Part (a): End Behavior Using Leading Coefficient Test
The leading term of the function is \( x^3 \). Since the degree (3) is odd and the leading coefficient (1) is positive, we can use the leading coefficient test:
- As \( x \to -\infty \), \( f(x) \to -\infty \) (the graph falls to the left).
- As \( x \to \infty \), \( f(x) \to \infty \) (the graph rises to the right).
Thus, the correct statement describing the end behavior of \( f(x) \) is:
**A. The graph falls to the left and rises to the right.**
### Part (b): Finding the x-intercepts
To find the x-intercepts, we set \( f(x) = 0 \):
\[
x^3 + 3x^2 - x - 3 = 0
\]
To find the roots, we can use synthetic division or try possible rational roots based on the Rational Root Theorem. We will test \( x = 1 \):
\[
f(1) = 1^3 + 3(1)^2 - 1 - 3 = 1 + 3 - 1 - 3 = 0
\]
So, \( x = 1 \) is a root. We can use synthetic division to divide the polynomial by \( x - 1 \):
```
1 | 1 3 -1 -3
| 1 4 3
-----------------
1 4 3 0
```
This gives us \( x^2 + 4x + 3 \). Now we can factor this quadratic:
\[
x^2 + 4x + 3 = (x + 1)(x + 3) = 0
\]
Thus, the roots of the original equation are:
\[
x = 1, x = -1, x = -3
\]
The x-intercepts are \( x = 1, -1, -3 \).
#### Crosses or Touches the x-axis
- Since all roots are simple (the multiplicity of each root is 1), the graph will cross the x-axis at each intercept.
Thus:
- **The x-intercepts are:**
\( x = 1, -1, -3 \) (write as \( 1, -1, -3 \))
- **At which x-intercept(s) does the graph cross the x-axis?**
A. **x = 1, -1, -3**
- **At which x-intercept(s) does the graph touch the x-axis and turn around?**
B. **There are no x-intercepts at which the graph touches the x-axis and turns around.**
### Summary of Answers:
- (a): A. The graph falls to the left and rises to the right.
- (b):
- The x-intercepts: \( 1, -1, -3 \)
- Crosses the x-axis at: A. \( 1, -1, -3 \)
- Touches the x-axis and turns around at: B. There are no such x-intercepts.
### Part (a): End Behavior Using Leading Coefficient Test
The leading term of the function is \( x^3 \). Since the degree (3) is odd and the leading coefficient (1) is positive, we can use the leading coefficient test:
- As \( x \to -\infty \), \( f(x) \to -\infty \) (the graph falls to the left).
- As \( x \to \infty \), \( f(x) \to \infty \) (the graph rises to the right).
Thus, the correct statement describing the end behavior of \( f(x) \) is:
**A. The graph falls to the left and rises to the right.**
### Part (b): Finding the x-intercepts
To find the x-intercepts, we set \( f(x) = 0 \):
\[
x^3 + 3x^2 - x - 3 = 0
\]
To find the roots, we can use synthetic division or try possible rational roots based on the Rational Root Theorem. We will test \( x = 1 \):
\[
f(1) = 1^3 + 3(1)^2 - 1 - 3 = 1 + 3 - 1 - 3 = 0
\]
So, \( x = 1 \) is a root. We can use synthetic division to divide the polynomial by \( x - 1 \):
```
1 | 1 3 -1 -3
| 1 4 3
-----------------
1 4 3 0
```
This gives us \( x^2 + 4x + 3 \). Now we can factor this quadratic:
\[
x^2 + 4x + 3 = (x + 1)(x + 3) = 0
\]
Thus, the roots of the original equation are:
\[
x = 1, x = -1, x = -3
\]
The x-intercepts are \( x = 1, -1, -3 \).
#### Crosses or Touches the x-axis
- Since all roots are simple (the multiplicity of each root is 1), the graph will cross the x-axis at each intercept.
Thus:
- **The x-intercepts are:**
\( x = 1, -1, -3 \) (write as \( 1, -1, -3 \))
- **At which x-intercept(s) does the graph cross the x-axis?**
A. **x = 1, -1, -3**
- **At which x-intercept(s) does the graph touch the x-axis and turn around?**
B. **There are no x-intercepts at which the graph touches the x-axis and turns around.**
### Summary of Answers:
- (a): A. The graph falls to the left and rises to the right.
- (b):
- The x-intercepts: \( 1, -1, -3 \)
- Crosses the x-axis at: A. \( 1, -1, -3 \)
- Touches the x-axis and turns around at: B. There are no such x-intercepts.