Problem 20 - Algebra 2

Question

Sketch the asymptotes and the graph of each equation. \(y=\frac{1}{x}+2\)

Step-by-Step Solution

Answer

The graph of the equation \(y=\frac{1}{x}+2\) is a hyperbola. It has a horizontal asymptote at y=2 and a vertical asymptote at x=0. The graph is approaching these asymptotes but never crosses them. For \(x > 0\), the graph lies above the horizontal asymptote, and for \(x < 0\), it lies below the horizontal asymptote.

Step 1: Identify and plot the asymptotes

Start by identifying the asymptotes. In this case, there is a horizontal asymptote at y = 2 and a vertical asymptote at x = 0. Plot these two lines on your graph.

Step 2: Plot key points

Next, plot key points on the graph, focusing on points to the right and left of the asymptotes. In this case, for \(x > 0\), as \(x\) approaches 0 from the right, \(y\) tends to infinity. As \(x\) tends to infinity, \(y\) tends to 2. Similarly, for \(x < 0\), as \(x\) tends to 0 from the left, \(y\) tends to negative infinity. As \(x\) tends to negative infinity, \(y\) tends to 2. So, plot the points (1,3), (2,2.5), (3,2.33), (-1,1), (-2,1.5), (-3,1.67) as an approximation.

Step 3: Sketch the graph

Lastly, draw the graph that goes smoothly through the plotted points and towards the asymptotes. For \(x > 0\), curve will be above and approaching the horizontal asymptote. For \(x < 0\), curve will be below and approaching the horizontal asymptote.

Key Concepts

Horizontal Asymptote

In the function \( y = \frac{1}{x} + 2 \), the horizontal asymptote is a horizontal line that the graph of the function approaches but never actually reaches. It provides insight into the end behavior of the graph as \( x \to \infty \) or \( x \to -\infty \). For this particular function, the horizontal asymptote is at \( y = 2 \). This occurs because as \( x \) becomes very large in magnitude, the fraction \( \frac{1}{x} \) tends to zero. Thus, the function \( y \approx 2 \) far from the origin.

• The horizontal asymptote tells us how the function behaves as \( x \) gets very large or very small.
• It does not affect the behavior of the function around \( x = 0 \), only at the extremes of the graph.

This asymptote is crucial when sketching the graph to know where the curve will level off.

Vertical Asymptote

For the expression \( y = \frac{1}{x} + 2 \), the vertical asymptote is located at \( x = 0 \). A vertical asymptote is a vertical line that the graph approaches but does not touch or cross. Vertical asymptotes usually occur where the function is undefined, such as when a denominator in a rational function equals zero.

• When \( x \) is close to 0 but greater than 0, \( y \rightarrow +\infty \).
• When \( x \) approaches 0 from the left, \( y \rightarrow -\infty \).

It is essential to graph this asymptote to understand the domain of the function and where the function sharply increases or decreases. This line divides the graph into two distinct parts, mirroring across \( y = 2 \) but remaining separate.

Plotting Key Points

Plotting key points helps visualize the graph's shape and confirm the asymptotic behavior. For \( y = \frac{1}{x} + 2 \), determine points that fall on either side of the vertical asymptote to show the function's growth or decay.

• For \( x > 0 \), pick points such as 1, 2, and 3.
• At \( x = 1 \), \( y = 3 \); at \( x = 2 \), \( y = 2.5 \); and at \( x = 3 \), \( y = 2.33 \).
• For \( x < 0 \), choose points like -1, -2, and -3.
• At \( x = -1 \), \( y = 1 \); at \( x = -2 \), \( y = 1.5 \); and at \( x = -3 \), \( y = 1.67 \).

Plotting these points gives a better understanding of how the graph curves upwards on the right and downwards on the left towards respective asymptotes.

Sketching Graphs

After identifying asymptotes and plotting key points, sketching the graph involves connecting the dots and noting the asymptotic behavior. Start by drawing smooth curves through the plotted points, ensuring they closely approach the horizontal and vertical asymptotes, but never intersect them.

• For \( x > 0 \), the graph approaches \( y = 2 \) from above.
• For \( x < 0 \), the graph approaches \( y = 2 \) from below.
• The portions of the graph should remain distinct due to the vertical asymptote at \( x = 0 \).

This step requires practice because it translates all the plotted information into an accurate depiction of the function. Remember, the key is to represent an approximation that follows the trends and patterns indicated by the asymptotes and plotted points.