Question
Follow the seven step strategy to graph the following rational function.
f(x)equalsStartFraction x Superscript 4 Over x squared plus 3 EndFraction
Question content area bottom
Part 1
1. Select the symmetry of the function.
y-axis symmetry
origin symmetry
neither y-axis symmetry nor origin symmetry
give graph, x and y intercept, horizontal and verticle asymptote
f(x)equalsStartFraction x Superscript 4 Over x squared plus 3 EndFraction
Question content area bottom
Part 1
1. Select the symmetry of the function.
y-axis symmetry
origin symmetry
neither y-axis symmetry nor origin symmetry
give graph, x and y intercept, horizontal and verticle asymptote
Answers
Answered by
GPT-4o mini
To analyze the rational function \( f(x) = \frac{x^4}{x^2 + 3} \) using the seven-step strategy, let's walk through each step.
### Step 1: Determine Symmetry
To check for symmetry:
1. **Y-axis symmetry**: Replace \( x \) with \( -x \).
\[
f(-x) = \frac{(-x)^4}{(-x)^2 + 3} = \frac{x^4}{x^2 + 3} = f(x)
\]
Since \( f(-x) = f(x) \), the function is symmetric about the y-axis.
So, the answer is **y-axis symmetry**.
### Step 2: Find the x-intercepts
To find the x-intercepts, set \( f(x) = 0 \):
\[
\frac{x^4}{x^2 + 3} = 0
\]
The numerator must equal zero:
\[
x^4 = 0 \implies x = 0
\]
Thus, the only x-intercept is at \( (0, 0) \).
### Step 3: Find the y-intercept
To find the y-intercept, evaluate \( f(0) \):
\[
f(0) = \frac{0^4}{0^2 + 3} = \frac{0}{3} = 0
\]
Thus, the y-intercept is also at \( (0, 0) \).
### Step 4: Find vertical asymptotes
Vertical asymptotes occur where the denominator is zero, provided the numerator is not zero at those points. Set the denominator to zero:
\[
x^2 + 3 = 0
\]
This equation has no real solutions since \( x^2 = -3 \) is not possible. Therefore, there are **no vertical asymptotes**.
### Step 5: Find horizontal asymptotes
To find horizontal asymptotes, analyze the behavior as \( x \) approaches infinity. Consider the degrees of the polynomial in the numerator and the denominator:
- Degree of the numerator: \( 4 \)
- Degree of the denominator: \( 2 \)
Since the degree of the numerator (4) is greater than the degree of the denominator (2), there is no horizontal asymptote. However, since the numerator grows faster than the denominator, we can say:
\[
\text{As } x \to \infty, f(x) \to \infty
\]
\[
\text{As } x \to -\infty, f(x) \to \infty
\]
### Step 6: Analyze end behavior
The end behavior further supports that as \( x \to \infty \) or \( x \to -\infty \), \( f(x) \) will increase without bound.
### Step 7: Graph the function
Now that we have all the information:
- **Symmetry**: y-axis symmetry
- **x-intercept**: (0, 0)
- **y-intercept**: (0, 0)
- **Vertical asymptotes**: none
- **Horizontal asymptotes**: none
Here’s what the graph will look like:
- The graph of \( f(x) = \frac{x^4}{x^2 + 3} \) will be a curve opening upward with a vertex at the origin. Since it's symmetric, it will be mirror image on both sides of the y-axis.
In a graphing tool or software, the graph would show a steep rise for large positive and negative values of \( x \).
### Summary
This function grows rapidly at both ends and has a single intercept at the origin. There are no asymptotes, which indicates the function is well-defined for all values of \( x \).
### Step 1: Determine Symmetry
To check for symmetry:
1. **Y-axis symmetry**: Replace \( x \) with \( -x \).
\[
f(-x) = \frac{(-x)^4}{(-x)^2 + 3} = \frac{x^4}{x^2 + 3} = f(x)
\]
Since \( f(-x) = f(x) \), the function is symmetric about the y-axis.
So, the answer is **y-axis symmetry**.
### Step 2: Find the x-intercepts
To find the x-intercepts, set \( f(x) = 0 \):
\[
\frac{x^4}{x^2 + 3} = 0
\]
The numerator must equal zero:
\[
x^4 = 0 \implies x = 0
\]
Thus, the only x-intercept is at \( (0, 0) \).
### Step 3: Find the y-intercept
To find the y-intercept, evaluate \( f(0) \):
\[
f(0) = \frac{0^4}{0^2 + 3} = \frac{0}{3} = 0
\]
Thus, the y-intercept is also at \( (0, 0) \).
### Step 4: Find vertical asymptotes
Vertical asymptotes occur where the denominator is zero, provided the numerator is not zero at those points. Set the denominator to zero:
\[
x^2 + 3 = 0
\]
This equation has no real solutions since \( x^2 = -3 \) is not possible. Therefore, there are **no vertical asymptotes**.
### Step 5: Find horizontal asymptotes
To find horizontal asymptotes, analyze the behavior as \( x \) approaches infinity. Consider the degrees of the polynomial in the numerator and the denominator:
- Degree of the numerator: \( 4 \)
- Degree of the denominator: \( 2 \)
Since the degree of the numerator (4) is greater than the degree of the denominator (2), there is no horizontal asymptote. However, since the numerator grows faster than the denominator, we can say:
\[
\text{As } x \to \infty, f(x) \to \infty
\]
\[
\text{As } x \to -\infty, f(x) \to \infty
\]
### Step 6: Analyze end behavior
The end behavior further supports that as \( x \to \infty \) or \( x \to -\infty \), \( f(x) \) will increase without bound.
### Step 7: Graph the function
Now that we have all the information:
- **Symmetry**: y-axis symmetry
- **x-intercept**: (0, 0)
- **y-intercept**: (0, 0)
- **Vertical asymptotes**: none
- **Horizontal asymptotes**: none
Here’s what the graph will look like:
- The graph of \( f(x) = \frac{x^4}{x^2 + 3} \) will be a curve opening upward with a vertex at the origin. Since it's symmetric, it will be mirror image on both sides of the y-axis.
In a graphing tool or software, the graph would show a steep rise for large positive and negative values of \( x \).
### Summary
This function grows rapidly at both ends and has a single intercept at the origin. There are no asymptotes, which indicates the function is well-defined for all values of \( x \).