Question
Use the given function to complete parts a) through e) below.
f left parenthesis x right parenthesis equals negative x Superscript 4 Baseline plus 25 x squared
Question content area bottom
Part 1
a) Use the Leading Coefficient Test to determine the graph's end behavior.
A.
The graph of f(x) rises left and rises right.
B.
The graph of f(x) falls left and falls right.
C.
The graph of f(x) falls left and rises right.
D.
The graph of f(x) rises left and falls right.
Part 2
b) Find the x-intercepts.
xequals
enter your response here
(Type an integer or a decimal. Use a comma to separate answers as needed.)
Part 3
At which zeros does the graph of the function 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 4
At which zeros does the graph of the function 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 5
c) Find the y-intercept by computing f(0).
f(0)equals
enter your response here
Part 6
d) Determine the symmetry of the graph.
Odd; origin symmetry
Even; y-axis symmetry
Neither
Part 7
e) Determine the graph of the function.
A.
-10
10
-200
200
x
y
The graph has a horizontal x-axis from negative 10 to 10 and a vertical y-axis from negative 200 to 200. The graph is a parabola passing through (negative 5,0) and (5,0). The graph falls to the left and falls to the right.
B.
-10
10
-200
200
x
y
The graph has a horizontal x-axis from negative 10 to 10 and a vertical y-axis from negative 200 to 200. The graph has a parabola opening downwards and passing through the origin.
C.
-10
10
-200
200
x
y
The graph has a horizontal x-axis from negative 10 to 10 and a vertical y-axis from negative 200 to 200. The graph falls to the left and falls to the right. From left to right the curve rises intersecting the x-axis at negative 5 and reaching a turning point. Then the curve falls to touch the x-axis at the origin. The curve rises to a turning point and then falls crossing the x-axis at 5.
D.
-10
10
-200
200
x
y
The graph has a horizontal x-axis from negative 10 to 10 and a vertical y-axis from negative 200 to 200. The graph is a U shaped curve opening downward and passing through the origin.
a) Use the Leading Coefficient Test to determine the graph's end behavior.x(Type an integer or a decimal. Use a comma to separate answers as needed.))Select the correct choice below and, if necessary, fill in the answer box to complete your choice.input field 3Select the correct choice below and, if necessary, fill in the answer box to complete your choice.f(0)d) Determine the symmetry of the graph.e) Determine the graph of the function.
f left parenthesis x right parenthesis equals negative x Superscript 4 Baseline plus 25 x squared
Question content area bottom
Part 1
a) Use the Leading Coefficient Test to determine the graph's end behavior.
A.
The graph of f(x) rises left and rises right.
B.
The graph of f(x) falls left and falls right.
C.
The graph of f(x) falls left and rises right.
D.
The graph of f(x) rises left and falls right.
Part 2
b) Find the x-intercepts.
xequals
enter your response here
(Type an integer or a decimal. Use a comma to separate answers as needed.)
Part 3
At which zeros does the graph of the function 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 4
At which zeros does the graph of the function 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 5
c) Find the y-intercept by computing f(0).
f(0)equals
enter your response here
Part 6
d) Determine the symmetry of the graph.
Odd; origin symmetry
Even; y-axis symmetry
Neither
Part 7
e) Determine the graph of the function.
A.
-10
10
-200
200
x
y
The graph has a horizontal x-axis from negative 10 to 10 and a vertical y-axis from negative 200 to 200. The graph is a parabola passing through (negative 5,0) and (5,0). The graph falls to the left and falls to the right.
B.
-10
10
-200
200
x
y
The graph has a horizontal x-axis from negative 10 to 10 and a vertical y-axis from negative 200 to 200. The graph has a parabola opening downwards and passing through the origin.
C.
-10
10
-200
200
x
y
The graph has a horizontal x-axis from negative 10 to 10 and a vertical y-axis from negative 200 to 200. The graph falls to the left and falls to the right. From left to right the curve rises intersecting the x-axis at negative 5 and reaching a turning point. Then the curve falls to touch the x-axis at the origin. The curve rises to a turning point and then falls crossing the x-axis at 5.
D.
-10
10
-200
200
x
y
The graph has a horizontal x-axis from negative 10 to 10 and a vertical y-axis from negative 200 to 200. The graph is a U shaped curve opening downward and passing through the origin.
a) Use the Leading Coefficient Test to determine the graph's end behavior.x(Type an integer or a decimal. Use a comma to separate answers as needed.))Select the correct choice below and, if necessary, fill in the answer box to complete your choice.input field 3Select the correct choice below and, if necessary, fill in the answer box to complete your choice.f(0)d) Determine the symmetry of the graph.e) Determine the graph of the function.
Answers
GPT-4o mini
To analyze the function \( f(x) = -x^4 + 25x^2 \), we will address each part of the question sequentially.
### Part 1: Leading Coefficient Test for End Behavior
1. **Degree of the polynomial**: The degree of \( f(x) \) is 4 (because of the \( x^4 \)).
2. **Leading coefficient**: The leading coefficient is -1 (which is negative).
- For even-degree polynomials:
- If the leading coefficient is positive, the graph rises to the left and rises to the right.
- If the leading coefficient is negative, the graph falls to the left and falls to the right.
**Conclusion**: The correct answer is **B. The graph of \( f(x) \) falls left and falls right.**
---
### Part 2: Finding the x-intercepts
To find the x-intercepts, set \( f(x) = 0 \):
\[
-x^4 + 25x^2 = 0
\]
Factor out the common term:
\[
-x^2(x^2 - 25) = 0
\]
This gives us:
\[
-x^2 = 0 \quad \Rightarrow \quad x = 0
\]
\[
x^2 - 25 = 0 \quad \Rightarrow \quad x^2 = 25 \quad \Rightarrow \quad x = 5 \text{ or } x = -5
\]
Thus, the x-intercepts are \( x = 0, 5, -5 \).
**Answer**: \( x = -5, 0, 5 \)
---
### Part 3: Crossing the x-axis
- The graph crosses the x-axis where the roots have **odd multiplicity**. Since \( x = 0 \) has even multiplicity (2 as a factor from \( -x^2 \)), it does not cross.
**Answer**: B. There are no x-intercepts at which the graph crosses the x-axis.
---
### Part 4: Touching the x-axis
- The graph touches the x-axis where the roots have **even multiplicity**. Here, both \( x = 0 \) (touches) and \( x = -5, 5 \) (cross).
**Answer**: A. \( x = 0 \)
---
### Part 5: Finding the y-intercept
To find the y-intercept, compute \( f(0) \):
\[
f(0) = -0^4 + 25 \cdot 0^2 = 0
\]
**Answer**: \( f(0) = 0 \)
---
### Part 6: Determining symmetry
- A function is **even** if \( f(-x) = f(x) \) for all x (y-axis symmetry).
- A function is **odd** if \( f(-x) = -f(x) \) for all x (origin symmetry).
Calculate \( f(-x) \):
\[
f(-x) = -(-x)^4 + 25(-x)^2 = -x^4 + 25x^2 = f(x)
\]
Since \( f(-x) = f(x) \), the function is even.
**Answer**: Even; y-axis symmetry.
---
### Part 7: Determine the graph of the function
Based on all the conclusions drawn:
- The graph falls to the left and falls to the right (end behavior).
- It has x-intercepts at \( -5, 0, 5 \), touches the x-axis at \( 0 \), and crosses the x-axis at \( 5 \) and \( -5 \).
**Answer**: The most appropriate description seems to align with **C**:
> The graph has a horizontal x-axis from -10 to 10 and a vertical y-axis from -200 to 200. The graph falls to the left and falls to the right. From left to right the curve rises intersecting the x-axis at negative 5 and reaching a turning point. Then the curve falls to touch the x-axis at the origin. The curve rises to a turning point and then falls crossing the x-axis at 5.
---
To summarize:
- Part 1: B
- Part 2: -5, 0, 5
- Part 3: B
- Part 4: A (0)
- Part 5: 0
- Part 6: Even; y-axis symmetry
- Part 7: C
### Part 1: Leading Coefficient Test for End Behavior
1. **Degree of the polynomial**: The degree of \( f(x) \) is 4 (because of the \( x^4 \)).
2. **Leading coefficient**: The leading coefficient is -1 (which is negative).
- For even-degree polynomials:
- If the leading coefficient is positive, the graph rises to the left and rises to the right.
- If the leading coefficient is negative, the graph falls to the left and falls to the right.
**Conclusion**: The correct answer is **B. The graph of \( f(x) \) falls left and falls right.**
---
### Part 2: Finding the x-intercepts
To find the x-intercepts, set \( f(x) = 0 \):
\[
-x^4 + 25x^2 = 0
\]
Factor out the common term:
\[
-x^2(x^2 - 25) = 0
\]
This gives us:
\[
-x^2 = 0 \quad \Rightarrow \quad x = 0
\]
\[
x^2 - 25 = 0 \quad \Rightarrow \quad x^2 = 25 \quad \Rightarrow \quad x = 5 \text{ or } x = -5
\]
Thus, the x-intercepts are \( x = 0, 5, -5 \).
**Answer**: \( x = -5, 0, 5 \)
---
### Part 3: Crossing the x-axis
- The graph crosses the x-axis where the roots have **odd multiplicity**. Since \( x = 0 \) has even multiplicity (2 as a factor from \( -x^2 \)), it does not cross.
**Answer**: B. There are no x-intercepts at which the graph crosses the x-axis.
---
### Part 4: Touching the x-axis
- The graph touches the x-axis where the roots have **even multiplicity**. Here, both \( x = 0 \) (touches) and \( x = -5, 5 \) (cross).
**Answer**: A. \( x = 0 \)
---
### Part 5: Finding the y-intercept
To find the y-intercept, compute \( f(0) \):
\[
f(0) = -0^4 + 25 \cdot 0^2 = 0
\]
**Answer**: \( f(0) = 0 \)
---
### Part 6: Determining symmetry
- A function is **even** if \( f(-x) = f(x) \) for all x (y-axis symmetry).
- A function is **odd** if \( f(-x) = -f(x) \) for all x (origin symmetry).
Calculate \( f(-x) \):
\[
f(-x) = -(-x)^4 + 25(-x)^2 = -x^4 + 25x^2 = f(x)
\]
Since \( f(-x) = f(x) \), the function is even.
**Answer**: Even; y-axis symmetry.
---
### Part 7: Determine the graph of the function
Based on all the conclusions drawn:
- The graph falls to the left and falls to the right (end behavior).
- It has x-intercepts at \( -5, 0, 5 \), touches the x-axis at \( 0 \), and crosses the x-axis at \( 5 \) and \( -5 \).
**Answer**: The most appropriate description seems to align with **C**:
> The graph has a horizontal x-axis from -10 to 10 and a vertical y-axis from -200 to 200. The graph falls to the left and falls to the right. From left to right the curve rises intersecting the x-axis at negative 5 and reaching a turning point. Then the curve falls to touch the x-axis at the origin. The curve rises to a turning point and then falls crossing the x-axis at 5.
---
To summarize:
- Part 1: B
- Part 2: -5, 0, 5
- Part 3: B
- Part 4: A (0)
- Part 5: 0
- Part 6: Even; y-axis symmetry
- Part 7: C