Question
Use the given function to complete parts a) through e) below.
f left parenthesis x right parenthesis equals negative 3 left parenthesis x minus 4 right parenthesis squared left parenthesis x squared minus 1 right parenthesis
Question content area bottom
Part 1
a) Use the Leading Coefficient Test to determine the graph's end behavior.
The graph of f(x) falls left and rises right.
The graph of f(x) falls left and falls right.
The graph of f(x) rises left and rises right.
The graph of f(x) rises left and falls right.
b. Find the x-intercept(s). State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept.
The x-intercept(s) is/are
enter your response here.
(Type an exact answer, using radicals as needed. Use a comma to separate answers as needed. Type each answer only once.)
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.
The x-intercept(s) at which the graph crosses the x-axis is/are
enter your response here.
(Type an exact answer, using radicals as needed. Use a comma to separate answers as needed. Type each answer only once.)
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.
The x-intercept(s) at which the graph touches the x-axis and turns around is/are
enter your response here.
(Type an exact answer, using radicals as needed. Use a comma to separate answers as needed. Type each answer only once.)
B.
There are no x-intercepts at which the graph touches the x-axis and turns around.
d. Determine whether the graph has y-axis symmetry, origin symmetry, or neither. Choose the correct answer below.
A.
The graph of f is symmetric about the origin.
B.
The graph of f is symmetric about the y-axis.
C.
The graph of f is neither symmetric about the y-axis nor symmetric about the origin.
e. If necessary, find a few additional points and graph the function. Use the maximum number of turning points to check whether it is drawn correctly. Choose the correct graph below.
f left parenthesis x right parenthesis equals negative 3 left parenthesis x minus 4 right parenthesis squared left parenthesis x squared minus 1 right parenthesis
Question content area bottom
Part 1
a) Use the Leading Coefficient Test to determine the graph's end behavior.
The graph of f(x) falls left and rises right.
The graph of f(x) falls left and falls right.
The graph of f(x) rises left and rises right.
The graph of f(x) rises left and falls right.
b. Find the x-intercept(s). State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept.
The x-intercept(s) is/are
enter your response here.
(Type an exact answer, using radicals as needed. Use a comma to separate answers as needed. Type each answer only once.)
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.
The x-intercept(s) at which the graph crosses the x-axis is/are
enter your response here.
(Type an exact answer, using radicals as needed. Use a comma to separate answers as needed. Type each answer only once.)
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.
The x-intercept(s) at which the graph touches the x-axis and turns around is/are
enter your response here.
(Type an exact answer, using radicals as needed. Use a comma to separate answers as needed. Type each answer only once.)
B.
There are no x-intercepts at which the graph touches the x-axis and turns around.
d. Determine whether the graph has y-axis symmetry, origin symmetry, or neither. Choose the correct answer below.
A.
The graph of f is symmetric about the origin.
B.
The graph of f is symmetric about the y-axis.
C.
The graph of f is neither symmetric about the y-axis nor symmetric about the origin.
e. If necessary, find a few additional points and graph the function. Use the maximum number of turning points to check whether it is drawn correctly. Choose the correct graph below.
Answers
GPT-4o mini
To analyze the function \( f(x) = -3(x - 4)^2(x^2 - 1) \) and answer the questions step by step, let’s go through each part:
### Part a: Leading Coefficient Test and End Behavior
The leading coefficient of \( f(x) \) is obtained from the term with the highest degree. When expanded, \( f(x) \) can be simplified to find its degree:
1. The term \( (x - 4)^2 \) expands to \( x^2 - 8x + 16 \) (degree 2).
2. The term \( (x^2 - 1) \) is already in standard form (degree 2).
When multiplied, the leading term produces:
\[
f(x) = -3(x^2)(x^2) = -3x^4 \quad \text{(degree 4 with a negative leading coefficient)}
\]
Using the Leading Coefficient Test:
- Since it's a fourth degree polynomial and the leading coefficient is negative, the end behavior is:
- Falls to the left (as \( x \to -\infty \))
- Falls to the right (as \( x \to \infty \))
So, the answer for part a is:
**The graph of \( f(x) \) falls left and falls right.**
### Part b: Finding the x-intercepts
To find the \( x \)-intercepts, we set \( f(x) = 0 \):
\[
-3(x - 4)^2(x^2 - 1) = 0
\]
This implies that either \( (x - 4)^2 = 0 \) or \( (x^2 - 1) = 0 \).
1. From \( (x - 4)^2 = 0 \):
- \( x - 4 = 0 \) implies \( x = 4 \) (multiplicity 2).
2. From \( (x^2 - 1) = 0 \):
- \( x^2 = 1 \) leads to \( x = 1 \) and \( x = -1 \) (each with multiplicity 1).
Thus, the \( x \)-intercepts are:
**The x-intercept(s) is/are \( -1, 1, 4 \).**
#### Checking if the graph crosses or touches the x-axis
- The \( x \)-intercept at \( x = 4 \) has even multiplicity (2), so the graph touches the x-axis and turns around.
- The \( x \)-intercepts at \( x = -1 \) and \( x = 1 \) have odd multiplicity (1), so the graph crosses the x-axis.
Thus:
- **At which \( x \)-intercept(s) does the graph cross the x-axis?**:
**\( -1, 1 \)**
- **At which \( x \)-intercept(s) does the graph touch the x-axis and turn around?**:
**\( 4 \)**
### Part d: Symmetry
To determine symmetry:
- For \( y \)-axis symmetry, we check if \( f(-x) = f(x) \).
- For origin symmetry, we check if \( f(-x) = -f(x) \).
Calculating \( f(-x) \):
\[
f(-x) = -3(-x - 4)^2((-x)^2 - 1) = -3(x + 4)^2(x^2 - 1)
\]
This does not equal \( f(x) \) and does not equal \( -f(x) \) either, which would confirm origin symmetry. Thus, there is neither symmetry.
The answer for part d is:
**C. The graph of \( f \) is neither symmetric about the y-axis nor symmetric about the origin.**
### Part e: Additional Points and Graphing
To graph \( f(x) \), it’s useful to find points around the \( x \)-intercepts, such as \( x = 0 \):
\[
f(0) = -3(0 - 4)^2(0^2 - 1) = -3(16)(-1) = 48
\]
Some important points could be:
- \( x = -2: f(-2) = -3(-6)^2(-3) = -3(36)(-3) = 324 \)
- \( x = 2: f(2) = -3(-2)^2(3) = -3(4)(3) = -36 \)
- \( x = 3: f(3) = -3(-1)^2(8) = -3(1)(8) = -24 \)
Using these points, you can sketch the graph:
- Starts high, decreases to \( -1 \) (cross), goes up to 4 (touch), and then falls again.
The graph would illustrate these behaviors and characteristics based on the calculated points. If specific graph options were provided, you would select the one that matches the characteristics derived above.
If you need further assistance with selecting the graph or confirming anything, feel free to ask!
### Part a: Leading Coefficient Test and End Behavior
The leading coefficient of \( f(x) \) is obtained from the term with the highest degree. When expanded, \( f(x) \) can be simplified to find its degree:
1. The term \( (x - 4)^2 \) expands to \( x^2 - 8x + 16 \) (degree 2).
2. The term \( (x^2 - 1) \) is already in standard form (degree 2).
When multiplied, the leading term produces:
\[
f(x) = -3(x^2)(x^2) = -3x^4 \quad \text{(degree 4 with a negative leading coefficient)}
\]
Using the Leading Coefficient Test:
- Since it's a fourth degree polynomial and the leading coefficient is negative, the end behavior is:
- Falls to the left (as \( x \to -\infty \))
- Falls to the right (as \( x \to \infty \))
So, the answer for part a is:
**The graph of \( f(x) \) falls left and falls right.**
### Part b: Finding the x-intercepts
To find the \( x \)-intercepts, we set \( f(x) = 0 \):
\[
-3(x - 4)^2(x^2 - 1) = 0
\]
This implies that either \( (x - 4)^2 = 0 \) or \( (x^2 - 1) = 0 \).
1. From \( (x - 4)^2 = 0 \):
- \( x - 4 = 0 \) implies \( x = 4 \) (multiplicity 2).
2. From \( (x^2 - 1) = 0 \):
- \( x^2 = 1 \) leads to \( x = 1 \) and \( x = -1 \) (each with multiplicity 1).
Thus, the \( x \)-intercepts are:
**The x-intercept(s) is/are \( -1, 1, 4 \).**
#### Checking if the graph crosses or touches the x-axis
- The \( x \)-intercept at \( x = 4 \) has even multiplicity (2), so the graph touches the x-axis and turns around.
- The \( x \)-intercepts at \( x = -1 \) and \( x = 1 \) have odd multiplicity (1), so the graph crosses the x-axis.
Thus:
- **At which \( x \)-intercept(s) does the graph cross the x-axis?**:
**\( -1, 1 \)**
- **At which \( x \)-intercept(s) does the graph touch the x-axis and turn around?**:
**\( 4 \)**
### Part d: Symmetry
To determine symmetry:
- For \( y \)-axis symmetry, we check if \( f(-x) = f(x) \).
- For origin symmetry, we check if \( f(-x) = -f(x) \).
Calculating \( f(-x) \):
\[
f(-x) = -3(-x - 4)^2((-x)^2 - 1) = -3(x + 4)^2(x^2 - 1)
\]
This does not equal \( f(x) \) and does not equal \( -f(x) \) either, which would confirm origin symmetry. Thus, there is neither symmetry.
The answer for part d is:
**C. The graph of \( f \) is neither symmetric about the y-axis nor symmetric about the origin.**
### Part e: Additional Points and Graphing
To graph \( f(x) \), it’s useful to find points around the \( x \)-intercepts, such as \( x = 0 \):
\[
f(0) = -3(0 - 4)^2(0^2 - 1) = -3(16)(-1) = 48
\]
Some important points could be:
- \( x = -2: f(-2) = -3(-6)^2(-3) = -3(36)(-3) = 324 \)
- \( x = 2: f(2) = -3(-2)^2(3) = -3(4)(3) = -36 \)
- \( x = 3: f(3) = -3(-1)^2(8) = -3(1)(8) = -24 \)
Using these points, you can sketch the graph:
- Starts high, decreases to \( -1 \) (cross), goes up to 4 (touch), and then falls again.
The graph would illustrate these behaviors and characteristics based on the calculated points. If specific graph options were provided, you would select the one that matches the characteristics derived above.
If you need further assistance with selecting the graph or confirming anything, feel free to ask!