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Asymptotes are essential when sketching graphs, they are lines that a graph approaches but never touches. In the context of the given equation, \(y = \frac{1}{x-1}\), there are two types of asymptotes to consider: vertical and horizontal. The **vertical asymptote** occurs where the denominator equals zero, making the function undefined. Here, that's at \(x=1\). So, a line is drawn at \(x=1\) which the graph of the function will approach but not cross. The **horizontal asymptote** pertains to the behavior of the function as \(x\) becomes very large or very small. For the equation \(y = \frac{1}{x-1}\), as \(x\) heads towards infinity or negative infinity, \(y\) approaches 0. This signifies a horizontal asymptote at \(y=0\).

• Vertical asymptote: \(x=1\)
• Horizontal asymptote: \(y=0\)

Understanding these asymptotes helps in anticipating the behavior of the graph as it approaches these lines.

Hyperbolic graphs have a specific and recognizable shape, often forming two separate curves that extend towards their asymptotes. For the equation \(y = \frac{1}{x-1}\), the graph represents a simple hyperbola. This means that as \(x\) approaches the vertical asymptote from the left, \(y\) will travel towards negative infinity, creating a curve in the quadrant with negative \(y\) values. Conversely, as you approach from the right, \(y\) will move towards positive infinity, forming a curve in the quadrant with positive \(y\) values. Such behavior is typical of **hyperbolic functions:** - They have two distinct branches. - The branches are symmetrical about their asymptotes. - Their curvature demonstrates rapidly changing values as \(x\) nears the vertical asymptote. By recognizing these features, students can sketch accurate graphs that precisely reflect the function's behavior.

**Point plotting** is a practical technique for graph sketching that involves calculating specific \(x\) and \(y\) coordinates and marking these points on a graph. It is especially helpful in illustrating the general shape and direction of a curve.For the function \(y = \frac{1}{x-1}\), key points can be identified around the vertical asymptote \(x=1\). - By evaluating the function at \(x=0\), \(y = \frac{1}{0-1} = -1\). So, plot the point \((0, -1)\). - For \(x=2\), \(y = \frac{1}{2-1} = 1\), giving another point \((2, 1)\). Plotting these points provides an outline of the graph's form and allows for a visual check against the expected hyperbolic nature.

• Choose \(x\) values close to the asymptotes to see changes clearly.
• Calculate corresponding \(y\) values and plot each point accurately.
• Use the points to connect and complete the curve, while considering the asymptotes.

With these steps, students create an accurate representation of the function’s graph.