Question
Follow the seven step strategy to graph the following rational function.
f left parenthesis x right parenthesis equals StartFraction 8 x Over x squared minus 4 EndFraction
Question content area bottom
Part 1
To graph the function, first determine the symmetry of the graph of f. Choose the correct answer below.
y-axis symmetry
neither y-axis symmetry nor origin symmetry
origin symmetry
Part 2
What is the y-intercept? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A.
The y-intercept is
enter your response here.
(Type an integer or a simplified fraction.)
B.
There is no y-intercept.
Part 3
What is/are the x-intercept(s)? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A.
The x-intercept(s) is
enter your response here.
(Type an integer or a simplified fraction. Use a comma to separate answers as needed.)
B.
There is no x-intercept.
Part 4
Find the vertical asymptote(s). Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A.
The equation(s) of the vertical asymptote(s) is/are
enter your response here.
(Type an equation. Use a comma to separate answers as needed.)
B.
There is no vertical asymptote.
Part 5
Find the horizontal asymptote(s). Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A.
The equation(s) of the horizontal asymptote(s) is/are
enter your response here.
(Type an equation. Use a comma to separate answers as needed.)
B.
There is no horizontal asymptote.
Part 6
Plot points between and beyond each x-intercept and vertical asymptote. Find the value of the function at the given value of x.
x
negative 6
negative 5
minus1
1
5
8
f left parenthesis x right parenthesis equals StartFraction 8 x Over x squared minus 4 EndFraction
enter your response here
enter your response here
enter your response here
enter your response here
enter your response here
enter your response here
(Simplify your answers.)
Part 7
Use the information obtained in the previous steps to graph the function between and beyond the vertical asymptotes. Choose the correct graph below.
A.
-10
10
-10
10
x
y
A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from negative 10 to 10 in increments of 1. The xy plane contains three solid branches and three dashed lines. The first dashed line is horizontal and intersects y-axis at 0. The second and third dashed lines are vertical and intersect the x-axis at negative 2 and 2 respectively. The left branch approaches x-axis, from above, as x approaches negative infinity, from the origin the middle branch approaches negative 2 from the right and from the origin the middle branch approaches 2 from the left, and the right branch approaches 0, from below, as x approaches positive infinity. The left branch approaches negative infinity as x approaches negative 2 from the left and the right branch approaches positive infinity as x approaches 2 from the right.
B.
-10
10
-10
10
x
y
A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from negative 10 to 10 in increments of 1. The xy plane contains one solid curve and two dashed lines. Both dashed lines are vertical and intersect the x-axis at negative 2 and 2 respectively. The solid curve is parabola that opens downward, having its vertex at origin and intersecting both vertical dashed lines.
C.
-10
10
-10
10
x
y
A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from negative 10 to 10 in increments of 1. The xy plane contains three solid branches and three dashed lines. The first dashed line is horizontal and intersects y-axis at 0. The second and third dashed lines are vertical and intersect the x-axis at negative 2 and 2 respectively. The left branch approaches x-axis, from below, as x approaches negative infinity, from the origin the middle branch approaches negative 2 from the right and from the origin the middle branch approaches 2 from the left, and the right branch approaches 0, from above, as x approaches positive infinity. The left branch approaches negative infinity as x approaches negative 2 from the left and the right branch approaches positive infinity as x approaches 2 from the right.
D.
-10
10
-10
10
x
y
A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from negative 10 to 10 in increments of 1. The xy plane contains one solid curve and two dashed lines. Both dashed lines are vertical and intersect the x-axis at negative 2 and 2 respectively. The solid curve is parabola that opens upward, having its vertex at origin and intersecting both vertical dashed lines.
Choose the correct answer below.input field 1Select the correct choice below and, if necessary, fill in the answer box to complete your choice.input field 2Select 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.input field 4Select the correct choice below and, if necessary, fill in the answer box to complete your choice.input field 5input field 6input field 7input field 8input field 9input field 10Choose the correct graph below.
f left parenthesis x right parenthesis equals StartFraction 8 x Over x squared minus 4 EndFraction
Question content area bottom
Part 1
To graph the function, first determine the symmetry of the graph of f. Choose the correct answer below.
y-axis symmetry
neither y-axis symmetry nor origin symmetry
origin symmetry
Part 2
What is the y-intercept? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A.
The y-intercept is
enter your response here.
(Type an integer or a simplified fraction.)
B.
There is no y-intercept.
Part 3
What is/are the x-intercept(s)? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A.
The x-intercept(s) is
enter your response here.
(Type an integer or a simplified fraction. Use a comma to separate answers as needed.)
B.
There is no x-intercept.
Part 4
Find the vertical asymptote(s). Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A.
The equation(s) of the vertical asymptote(s) is/are
enter your response here.
(Type an equation. Use a comma to separate answers as needed.)
B.
There is no vertical asymptote.
Part 5
Find the horizontal asymptote(s). Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A.
The equation(s) of the horizontal asymptote(s) is/are
enter your response here.
(Type an equation. Use a comma to separate answers as needed.)
B.
There is no horizontal asymptote.
Part 6
Plot points between and beyond each x-intercept and vertical asymptote. Find the value of the function at the given value of x.
x
negative 6
negative 5
minus1
1
5
8
f left parenthesis x right parenthesis equals StartFraction 8 x Over x squared minus 4 EndFraction
enter your response here
enter your response here
enter your response here
enter your response here
enter your response here
enter your response here
(Simplify your answers.)
Part 7
Use the information obtained in the previous steps to graph the function between and beyond the vertical asymptotes. Choose the correct graph below.
A.
-10
10
-10
10
x
y
A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from negative 10 to 10 in increments of 1. The xy plane contains three solid branches and three dashed lines. The first dashed line is horizontal and intersects y-axis at 0. The second and third dashed lines are vertical and intersect the x-axis at negative 2 and 2 respectively. The left branch approaches x-axis, from above, as x approaches negative infinity, from the origin the middle branch approaches negative 2 from the right and from the origin the middle branch approaches 2 from the left, and the right branch approaches 0, from below, as x approaches positive infinity. The left branch approaches negative infinity as x approaches negative 2 from the left and the right branch approaches positive infinity as x approaches 2 from the right.
B.
-10
10
-10
10
x
y
A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from negative 10 to 10 in increments of 1. The xy plane contains one solid curve and two dashed lines. Both dashed lines are vertical and intersect the x-axis at negative 2 and 2 respectively. The solid curve is parabola that opens downward, having its vertex at origin and intersecting both vertical dashed lines.
C.
-10
10
-10
10
x
y
A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from negative 10 to 10 in increments of 1. The xy plane contains three solid branches and three dashed lines. The first dashed line is horizontal and intersects y-axis at 0. The second and third dashed lines are vertical and intersect the x-axis at negative 2 and 2 respectively. The left branch approaches x-axis, from below, as x approaches negative infinity, from the origin the middle branch approaches negative 2 from the right and from the origin the middle branch approaches 2 from the left, and the right branch approaches 0, from above, as x approaches positive infinity. The left branch approaches negative infinity as x approaches negative 2 from the left and the right branch approaches positive infinity as x approaches 2 from the right.
D.
-10
10
-10
10
x
y
A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 1 and a vertical y-axis labeled from negative 10 to 10 in increments of 1. The xy plane contains one solid curve and two dashed lines. Both dashed lines are vertical and intersect the x-axis at negative 2 and 2 respectively. The solid curve is parabola that opens upward, having its vertex at origin and intersecting both vertical dashed lines.
Choose the correct answer below.input field 1Select the correct choice below and, if necessary, fill in the answer box to complete your choice.input field 2Select 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.input field 4Select the correct choice below and, if necessary, fill in the answer box to complete your choice.input field 5input field 6input field 7input field 8input field 9input field 10Choose the correct graph below.
Answers
Answered by
GPT-4o mini
To analyze the rational function \( f(x) = \frac{8x}{x^2 - 4} \), we'll follow the seven-step strategy as outlined in the problem.
### Part 1: Determine the Symmetry
To find symmetry, we check:
- **y-axis symmetry**: \( f(-x) = f(x) \).
- **origin symmetry**: \( f(-x) = -f(x) \).
Calculating:
\[ f(-x) = \frac{8(-x)}{(-x)^2 - 4} = \frac{-8x}{x^2 - 4} = -\frac{8x}{x^2 - 4} = -f(x) \]
Thus, the function has **origin symmetry**.
**Answer: Origin symmetry.**
---
### Part 2: Find y-intercept
The y-intercept occurs when \( x = 0 \):
\[ f(0) = \frac{8(0)}{0^2 - 4} = \frac{0}{-4} = 0 \]
Thus, the y-intercept is 0.
**Answer: A. The y-intercept is 0.**
---
### Part 3: Find x-intercepts
The x-intercepts occur when \( f(x) = 0 \):
\[ \frac{8x}{x^2 - 4} = 0 \Rightarrow 8x = 0 \Rightarrow x = 0 \]
Thus, the x-intercept is also 0.
**Answer: A. The x-intercepts is 0.**
---
### Part 4: Find Vertical Asymptotes
Vertical asymptotes occur where the denominator is zero:
\[ x^2 - 4 = 0 \Rightarrow (x - 2)(x + 2) = 0 \Rightarrow x = 2 \text{ or } x = -2 \]
Thus, the vertical asymptotes are \( x = -2 \) and \( x = 2 \).
**Answer: A. The equations of the vertical asymptotes are \( x = -2, 2 \).**
---
### Part 5: Find Horizontal Asymptotes
For rational functions, the horizontal asymptote can be found by considering the degrees of the numerator and denominator. Here:
- The degree of the numerator (8x) is 1.
- The degree of the denominator (\(x^2 - 4\)) is 2.
Since the degree of the denominator is greater, the horizontal asymptote is:
\[ y = 0. \]
**Answer: A. The equation of the horizontal asymptote is \( y = 0 \).**
---
### Part 6: Evaluate the Function at Given x-values
We will calculate \( f(x) \) for \( x = -6, -5, -1, 1, 5, 8 \):
1. \( f(-6) = \frac{8(-6)}{(-6)^2 - 4} = \frac{-48}{36 - 4} = \frac{-48}{32} = -\frac{3}{2} \)
2. \( f(-5) = \frac{8(-5)}{(-5)^2 - 4} = \frac{-40}{25 - 4} = \frac{-40}{21} \)
3. \( f(-1) = \frac{8(-1)}{(-1)^2 - 4} = \frac{-8}{1 - 4} = \frac{-8}{-3} = \frac{8}{3} \)
4. \( f(1) = \frac{8(1)}{(1)^2 - 4} = \frac{8}{1 - 4} = \frac{8}{-3} = -\frac{8}{3} \)
5. \( f(5) = \frac{8(5)}{(5)^2 - 4} = \frac{40}{25 - 4} = \frac{40}{21} \)
6. \( f(8) = \frac{8(8)}{(8)^2 - 4} = \frac{64}{64 - 4} = \frac{64}{60} = \frac{16}{15} \)
Values obtained:
- \( f(-6) = -\frac{3}{2} \)
- \( f(-5) = -\frac{40}{21} \)
- \( f(-1) = \frac{8}{3} \)
- \( f(1) = -\frac{8}{3} \)
- \( f(5) = \frac{40}{21} \)
- \( f(8) = \frac{16}{15} \)
**Final values for part 6:**
- \( f(-6) = -\frac{3}{2} \)
- \( f(-5) = -\frac{40}{21} \)
- \( f(-1) = \frac{8}{3} \)
- \( f(1) = -\frac{8}{3} \)
- \( f(5) = \frac{40}{21} \)
- \( f(8) = \frac{16}{15} \)
---
### Part 7: Choose the Correct Graph
Based on the findings:
1. Origin symmetry.
2. x-intercept and y-intercept at \( (0, 0) \).
3. Vertical asymptotes at \( x = -2 \) and \( x = 2 \).
4. Horizontal asymptote at \( y = 0 \).
5. The behavior near asymptotes.
Given these characteristics, the graph must show these key features. This description matches choice **A** based on the outline of the branches and asymptotes described.
**Final Answer: The correct graph is A.**
### Part 1: Determine the Symmetry
To find symmetry, we check:
- **y-axis symmetry**: \( f(-x) = f(x) \).
- **origin symmetry**: \( f(-x) = -f(x) \).
Calculating:
\[ f(-x) = \frac{8(-x)}{(-x)^2 - 4} = \frac{-8x}{x^2 - 4} = -\frac{8x}{x^2 - 4} = -f(x) \]
Thus, the function has **origin symmetry**.
**Answer: Origin symmetry.**
---
### Part 2: Find y-intercept
The y-intercept occurs when \( x = 0 \):
\[ f(0) = \frac{8(0)}{0^2 - 4} = \frac{0}{-4} = 0 \]
Thus, the y-intercept is 0.
**Answer: A. The y-intercept is 0.**
---
### Part 3: Find x-intercepts
The x-intercepts occur when \( f(x) = 0 \):
\[ \frac{8x}{x^2 - 4} = 0 \Rightarrow 8x = 0 \Rightarrow x = 0 \]
Thus, the x-intercept is also 0.
**Answer: A. The x-intercepts is 0.**
---
### Part 4: Find Vertical Asymptotes
Vertical asymptotes occur where the denominator is zero:
\[ x^2 - 4 = 0 \Rightarrow (x - 2)(x + 2) = 0 \Rightarrow x = 2 \text{ or } x = -2 \]
Thus, the vertical asymptotes are \( x = -2 \) and \( x = 2 \).
**Answer: A. The equations of the vertical asymptotes are \( x = -2, 2 \).**
---
### Part 5: Find Horizontal Asymptotes
For rational functions, the horizontal asymptote can be found by considering the degrees of the numerator and denominator. Here:
- The degree of the numerator (8x) is 1.
- The degree of the denominator (\(x^2 - 4\)) is 2.
Since the degree of the denominator is greater, the horizontal asymptote is:
\[ y = 0. \]
**Answer: A. The equation of the horizontal asymptote is \( y = 0 \).**
---
### Part 6: Evaluate the Function at Given x-values
We will calculate \( f(x) \) for \( x = -6, -5, -1, 1, 5, 8 \):
1. \( f(-6) = \frac{8(-6)}{(-6)^2 - 4} = \frac{-48}{36 - 4} = \frac{-48}{32} = -\frac{3}{2} \)
2. \( f(-5) = \frac{8(-5)}{(-5)^2 - 4} = \frac{-40}{25 - 4} = \frac{-40}{21} \)
3. \( f(-1) = \frac{8(-1)}{(-1)^2 - 4} = \frac{-8}{1 - 4} = \frac{-8}{-3} = \frac{8}{3} \)
4. \( f(1) = \frac{8(1)}{(1)^2 - 4} = \frac{8}{1 - 4} = \frac{8}{-3} = -\frac{8}{3} \)
5. \( f(5) = \frac{8(5)}{(5)^2 - 4} = \frac{40}{25 - 4} = \frac{40}{21} \)
6. \( f(8) = \frac{8(8)}{(8)^2 - 4} = \frac{64}{64 - 4} = \frac{64}{60} = \frac{16}{15} \)
Values obtained:
- \( f(-6) = -\frac{3}{2} \)
- \( f(-5) = -\frac{40}{21} \)
- \( f(-1) = \frac{8}{3} \)
- \( f(1) = -\frac{8}{3} \)
- \( f(5) = \frac{40}{21} \)
- \( f(8) = \frac{16}{15} \)
**Final values for part 6:**
- \( f(-6) = -\frac{3}{2} \)
- \( f(-5) = -\frac{40}{21} \)
- \( f(-1) = \frac{8}{3} \)
- \( f(1) = -\frac{8}{3} \)
- \( f(5) = \frac{40}{21} \)
- \( f(8) = \frac{16}{15} \)
---
### Part 7: Choose the Correct Graph
Based on the findings:
1. Origin symmetry.
2. x-intercept and y-intercept at \( (0, 0) \).
3. Vertical asymptotes at \( x = -2 \) and \( x = 2 \).
4. Horizontal asymptote at \( y = 0 \).
5. The behavior near asymptotes.
Given these characteristics, the graph must show these key features. This description matches choice **A** based on the outline of the branches and asymptotes described.
**Final Answer: The correct graph is A.**