Problem 54
Question
Sketch the graph of the rational function by hand. As sketching aids, check for intercepts, vertical asymptotes, and slant asymptotes. $$f(x)=\frac{x^{3}}{x^{2}+4}$$
Step-by-Step Solution
Verified Answer
The graph of \( f(x)=\frac{x^{3}}{x^{2}+4} \) has one intercept at \(x = 0, y = 0\), no vertical asymptotes, and a slant asymptote at \(y = x\).
1Step 1: Find the Intercepts
To find the intercepts, set \(f(x)\) equal to zero and solve for \(x\). The function \(f(x)\) is zero when the numerator \(x^3\) is zero, so there is only one zero, \(x = 0\). Therefore, the intercepts of the function are at \(x = 0\) and \(y = 0\).
2Step 2: Determine Vertical Asymptotes
Vertical asymptotes occur where the denominator of the function is zero, so set \(x^2 + 4 = 0\) and solve for \(x\). There are no real solutions. Hence, the function has no vertical asymptotes.
3Step 3: Determine Slant Asymptotes
Note that the function will have a slant or oblique asymptote because the degree of the numerator is one greater than the degree of the denominator. Perform the division of the numerator by the denominator using polynomial division. The quotient becomes the equation of the slant asymptote. In this case, the quotient is \(x\), hence the equation of the slant asymptote is \(y = x\).
Key Concepts
InterceptsAsymptotesPolynomial Division
Intercepts
In rational functions, intercepts are crucial in understanding the graph's behavior. To find the intercepts of a rational function, we determine where the graph crosses the axes.
For the provided function, the x-intercept occurs where the numerator is zero, as this makes the whole function zero. Since the numerator is the cubic term \(x^3\), the only solution is \(x = 0\). This gives the x-intercept at (0,0), which is also the point where the line crosses the y-axis, making it the y-intercept as well.
To summarize on finding intercepts:
For the provided function, the x-intercept occurs where the numerator is zero, as this makes the whole function zero. Since the numerator is the cubic term \(x^3\), the only solution is \(x = 0\). This gives the x-intercept at (0,0), which is also the point where the line crosses the y-axis, making it the y-intercept as well.
To summarize on finding intercepts:
- X-intercepts: Set the numerator equal to zero and solve for \(x\).
- Y-intercepts: Set \(x=0\) and solve for \(f(x)\).
Asymptotes
Asymptotes are lines that the graph of a function approaches but never actually touches. There are several types, including vertical and slant asymptotes. For rational functions, understanding these can significantly aid graph sketching.
In the given function, there are no vertical asymptotes. They usually occur where the denominator is zero, but since the expression \(x^2 + 4\) has no real solutions (as squares of numbers plus a positive number can't equal zero), no vertical asymptotes exist.
However, the function does have a slant (oblique) asymptote. This occurs when the degree of the numerator is exactly one greater than the denominator's degree. Through polynomial division, we find that the slant asymptote is \(y = x\). This acts as a guiding line that the graph approaches at infinity. Thus,
In the given function, there are no vertical asymptotes. They usually occur where the denominator is zero, but since the expression \(x^2 + 4\) has no real solutions (as squares of numbers plus a positive number can't equal zero), no vertical asymptotes exist.
However, the function does have a slant (oblique) asymptote. This occurs when the degree of the numerator is exactly one greater than the denominator's degree. Through polynomial division, we find that the slant asymptote is \(y = x\). This acts as a guiding line that the graph approaches at infinity. Thus,
- Vertical Asymptote: None because \(x^2 + 4\) has no real roots.
- Slant Asymptote: Use polynomial division when the degree of the numerator is higher by one, resulting in \(y = x\).
Polynomial Division
Polynomial division is an essential tool when working with rational functions, particularly when finding slant asymptotes. It is akin to long division but with polynomials.
In our function, the polynomial division is required because the degree of the numerator \(x^3\) is greater than the degree of the denominator \(x^2 + 4\). By dividing these polynomials, we determine the slant asymptote's equation. Start by dividing \(x^3\) by \(x^2\) to obtain the first term \(x\). This process continues, but since we are concerned only with the first quotient for the asymptote, we find \(y = x\).
In summary, polynomial division is employed to:
In our function, the polynomial division is required because the degree of the numerator \(x^3\) is greater than the degree of the denominator \(x^2 + 4\). By dividing these polynomials, we determine the slant asymptote's equation. Start by dividing \(x^3\) by \(x^2\) to obtain the first term \(x\). This process continues, but since we are concerned only with the first quotient for the asymptote, we find \(y = x\).
In summary, polynomial division is employed to:
- Determine which type of asymptote exists.
- Find the equation of a slant asymptote when the numerator's degree is greater by one than the denominator's degree.
Other exercises in this chapter
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