Problem 49 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch the graph of the rational function by hand. As sketching aids, check for intercepts, vertical asymptotes, and slant asymptotes. $$f(x)=\frac{2 x^{2}+1}{x}$$

Step-by-Step Solution

Answer

The function \(f(x) = \frac{2 x^{2} + 1}{x}\) has a vertical asymptote at \(x = 0\) and a slant asymptote at \(y = 2x\). This function has no intercepts. The graph of the function can be sketched using these elements, plotting some function values and connecting the points smoothly, making sure the function approaches the asymptotes.

Step 1: Identify the Vertical Asymptote(s)

The vertical asymptote(s) of a rational function are found by setting the denominator equal to zero and solving for \(x\). For the given function \(f(x)=\frac{2 x^{2}+1}{x}\), the denominator is \(x\). So, setting \(x=0\) gives us the vertical asymptote \(x=0\).

Step 2: Identify the Horizontal/Slant Asymptote

For a rational function, if the degree of the numerator is greater than the degree of the denominator, the function has a slant asymptote. The equation of the slant asymptote is obtained by performing the long division of the numerator by the denominator. In this case, the degree of the numerator (which is 2) is greater than the degree of the denominator (which is 1), so there is a slant asymptote. When we divide \( 2x^2 + 1 \) by \( x \), we obtain \(2x\) as the quotient. Hence, the slant asymptote is \( y = 2x \).

Step 3: Identify the zero(s) of the function

The zeros of the function are found by setting the numerator equal to zero and solving for \(x\). In this case, the equation \(2x^2 + 1 = 0\) does not have any real roots, hence the function has no zeros.

Step 4: Find the y-intercept

The y-intercept is the point where the graph intersects the y-axis. This occurs when \(x=0\). But since \(x=0\) is the vertical asymptote, the graph does not intersect the y-axis, hence there is no y-intercept.

Step 5: Sketch the graph

Using the above information, plot the vertical asymptote as a dotted line at \(x=0\), and the slant asymptote as a dotted line with the equation \(y=2x\). Since there are no intercepts, plot points for several values of x on either side of the vertical asymptote and draw the graph accordingly. The graph will approach but never touch the asymptotes.

Key Concepts

Vertical Asymptotes

In rational functions, vertical asymptotes display the behavior of the graph as it moves towards specific, undefined values of the function. They occur when the denominator equals zero, creating a division by zero, which is undefined in mathematics.
For the given function \( f(x) = \frac{2x^2 + 1}{x} \), the vertical asymptote is found by setting the denominator \( x \) equal to zero.
This yields \( x = 0 \) as a vertical asymptote.

• Remember, a graph will approach a vertical asymptote but never actually touch or cross it.
• Visually, this is presented as a vertical dashed line across the graph at \( x = 0 \).
• Such lines help us understand restrictions on the domain of the function.

Understanding vertical asymptotes is crucial, as they depict points where the function heads towards infinity, either positively or negatively.
Be aware that in real-world applications, these indicate crucial data points or constraints.

Slant Asymptotes

Slant asymptotes, also known as oblique asymptotes, occur in rational functions when the degree of the numerator is exactly one more than the degree of the denominator.
In the function \( f(x) = \frac{2x^2 + 1}{x} \), the degree of the numerator is 2, while the denominator is 1, indicating a slant asymptote.
To determine the slant asymptote, perform a polynomial long division of the numerator by the denominator:

• Divide \( 2x^2 + 1 \) by \( x \).
• You'll find the quotient is \( 2x \), meaning the slant asymptote is represented by the line \( y = 2x \).

As graphs of functions approach a slant asymptote, they mimic the slant line at greater magnitude of \( x \). This interaction highlights the end behavior of the function, suggesting that as \( x \) moves towards positive or negative infinity, the function behaves more like the linear equation \( y = 2x \).
Slant asymptotes bring a fascinating dimension to graphing, offering insights not only into function characteristics but also into potential limits in certain real-life scenarios.

Intercepts

Intercepts are vital in graph analysis and provide insights into points where the graph crosses the axes, merging algebra with geometry. However, this particular function presents some unique circumstances.
For the rational function \( f(x) = \frac{2x^2 + 1}{x} \):
1. **X-Intercept:** The x-intercepts occur where the numerator is zero:

• Setting \( 2x^2 + 1 = 0 \), however, produces no real solutions. This tells us there are no x-intercepts, as the graph does not cross the x-axis.

2. **Y-Intercept:** The y-intercept is identified at \( x = 0 \).

• But here, \( x = 0 \) is where our vertical asymptote lies, meaning the function cannot intersect the y-axis.

Intercepts typically reveal critical points where functions relate back to their basic equation forms at axis crossings. Even in cases where intercepts are "missing," like here, it helps understand the broader topology of the graph.
Comprehending intercepts and their absence deepens insight into rational functions, aiding in crafting accurate and informative graphical representations.