Problem 64 - Algebra for College Students

Question

Graph the equation \(y=\frac{1}{x}\) (Example 4) using the following boundaries. (a) \(-15 \leq x \leq 15\) and \(-10 \leq y \leq 10\) (b) \(-10 \leq x \leq 10\) and \(-10 \leq y \leq 10\) (c) \(-5 \leq x \leq 5\) and \(-5 \leq y \leq 5\)

Step-by-Step Solution

Answer

Graph the hyperbola y = 1/x for each specified boundary, ensuring it approaches the axes asymptotically.

Step 1: Understanding the Equation

The equation we are working with is \( y = \frac{1}{x} \). This is a hyperbola that has two branches, one in each quadrant alternatively, typically in the first and third quadrants for a positive \( k \) value. The graph never touches the axes (asymptotes) and approaches them infinitely.

Step 2: Setting the Boundaries

For each case, we will graph the function within specified boundaries on both the x and y axes. We need to ensure the graph remains within these boundaries but shows the general form of the function. The boundaries are: (a) \(-15 \leq x \leq 15\) and \(-10 \leq y \leq 10\), (b) \(-10 \leq x \leq 10\) and \(-10 \leq y \leq 10\), (c) \(-5 \leq x \leq 5\) and \(-5 \leq y \leq 5\).

Step 3: Graph for Boundary (a)

For \( -15 \leq x \leq 15 \) and \( -10 \leq y \leq 10 \), plot points by choosing several x-values in this range and compute their corresponding y-values using \( y = \frac{1}{x} \). Plot these pairs, ensuring the graph approaches the y-axis and x-axis but does not touch them.

Step 4: Graph for Boundary (b)

Similarly, for \( -10 \leq x \leq 10 \) and \( -10 \leq y \leq 10 \), calculate y-values for x-values within this smaller range. The shape of the hyperbola will remain the same, but it will be narrower due to the reduced range.

Step 5: Graph for Boundary (c)

For \( -5 \leq x \leq 5 \) and \( -5 \leq y \leq 5 \), further restrict the x-values and thus the resulting y-values based on \( y = \frac{1}{x} \). The graph should depict the hyperbola's steep sections more clearly, focusing near the origin, demonstrating the function's typical characteristics.

Key Concepts

Asymptotes

Asymptotes are invisible lines on a graph that a function approaches but never actually reaches or crosses. In the case of the equation \(y = \frac{1}{x}\), there are two primary asymptotes to consider: the x-axis and y-axis. These are also known as the horizontal and vertical asymptotes, respectively. The graph of the hyperbola will come infinitely close to these axes, but it will never touch them.

• Horizontal asymptote: In \(y = \frac{1}{x}\), as \(x\) becomes very large (positively or negatively), \(y\) approaches 0. Thus, the x-axis (\(y=0\)) is the horizontal asymptote.
• Vertical asymptote: As \(x\) approaches 0, the values of \(y\) increase or decrease without bound. Therefore, the y-axis (\(x=0\)) acts as the vertical asymptote.

Understanding asymptotes is crucial because they guide the shape of the hyperbola. The graph will always bend towards these lines in an attempt to reach them but will never intersect.

Function Graphing

Graphing functions such as hyperbolas involves plotting points and sketching curves according to set mathematical rules. When graphing \(y = \frac{1}{x}\), there are two branches usually located in the first and third quadrants of the coordinate system.To graph this particular hyperbola, follow these steps:

• Select a range of x-values within the given boundaries, for example, from -15 to 15.
• Calculate the corresponding y-values using \(y = \frac{1}{x}\).
• Plot each (x, y) pair on the graph.
• Sketch smooth curves through the points, ensuring the graph bends towards the asymptotes but doesn't touch them.

Graphing functions accurately helps visualize relationships and understand the properties of equations. It's essential to follow the x and y boundaries, as recommending in the exercise, to ensure a clear and focused graph.

Coordinate System

The coordinate system is a foundational concept in graphing that allows us to represent mathematical equations visually. It consists of two perpendicular axes: the horizontal x-axis and the vertical y-axis.

• X-axis: This horizontal line helps determine the horizontal position of points on a graph.
• Y-axis: This vertical line helps determine the vertical position of points on a graph.

When graphing \(y = \frac{1}{x}\), the coordinate system provides a framework to plot the hyperbola accurately. The specified boundaries, like \(-15 \leq x \leq 15\) and \(-10 \leq y \leq 10\), delineate the viewing "window," which keeps the graph focused within these limits. This helps you better analyze the graph's behavior within a particular region.The system is a powerful tool, making it easier to understand the behavior and the relationships between variables in an equation. Every point on the graph is defined by a pair \((x, y)\), which expresses its exact position in this system.