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Magazine Chapter 1: Problem 9
Graph the equation after determining the \(x\) - and \(y\) -intercepts and whether the graph possesses any of the three types of symmetry described on page 58 $$y=-1 / x$$
Short Answer
Expert verified
The graph has no x- or y-intercepts and shows origin symmetry.
Step by step solution
01
Determining the X-Intercept
To find the x-intercept, we set \(y = 0\) and solve the equation \(0 = -1/x\). Since there is no real number \(x\) that will satisfy this equation, the graph has no x-intercept.
02
Determining the Y-Intercept
To find the y-intercept, set \(x = 0\) in the equation \(y = -1/x\). However, division by zero is undefined, indicating that there is no y-intercept for this equation.
03
Analyzing Symmetry with Respect to the Y-Axis
For y-axis symmetry, substituting \(-x\) for \(x\) in \(y = -1/x\) gives \(y = -1/(-x) = 1/x\), which is not equal to the original equation. Thus, the graph is not symmetric with respect to the y-axis.
04
Analyzing Symmetry with Respect to the X-Axis
For x-axis symmetry, substitute \(-y\) for \(y\). This gives \(-y = -1/x\) or \(y = 1/x\), which again is not equal to the original equation. Thus, the graph is not symmetric with respect to the x-axis.
05
Analyzing Symmetry with Respect to the Origin
For origin symmetry, substitute \(-x\) for \(x\) and \(-y\) for \(y\) in the equation. We have \(-y = -1/(-x)\) or \(y = 1/x = -(-1/x) = -1/x\), which is equal to the original equation. Thus, the graph has origin symmetry.
06
Graphing the Equation
Without intercepts, we sketch the graph using known points. The equation \(y = -1/x\) creates a hyperbola with asymptotes at the axes. Approach negative and positive infinity along both the x and y-axes. Note symmetric behavior around the origin.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding X-Intercepts
X-Intercepts are points where a graph crosses the x-axis. For a rational function like the one in the given exercise, you find them by setting the y-value to zero and solving for x. In the equation \( y = -1/x \), setting \( y = 0 \) leads to \( 0 = -1/x \), which is an impossible equation because you cannot divide 1 by any real number to get zero. Hence, this rational function has no x-intercepts. Understanding why an x-intercept exists or not helps you grasp the behavior and shape of the graph as it interacts with the x-axis.
Investigating Y-Intercepts
Y-Intercepts are where the graph meets the y-axis. To find them for a rational function, set \( x = 0 \). In our example, substituting \( x = 0 \) into \( y = -1/x \) results in division by zero. This is undefined in mathematics, meaning the graph of \( y = -1/x \) does not cross the y-axis and thus has no y-intercepts. Understanding y-intercepts helps us anticipate how and where the graph is situated in relation to the y-axis, providing insight into not only intercepts but also the domain of the function.
Conducting Symmetry Analysis
Symmetry analysis tells us if a graph looks the same from different perspectives. There are three types to consider: symmetry with respect to the y-axis, x-axis, and the origin. In our equation \( y = -1/x \):
- Y-Axis Symmetry: Replacing \( x \) with \( -x \) gives a different equation \( y = 1/x \). Thus, it's not symmetric concerning the y-axis.
- X-Axis Symmetry: Replacing \( y \) with \( -y \) yields \( y = 1/x \), indicating it鈥檚 not x-axis symmetric.
- Origin Symmetry: Substituting both \( -x \) and \( -y \) results in the original function, confirming symmetry about the origin.
Exploring Asymptotes
Asymptotes are lines that a graph approaches but never actually meets. They can be vertical, horizontal, or oblique. In the case of \( y = -1/x \), you need to identify:
- Vertical Asymptote: Occurs when the denominator is zero, \( x = 0 \), making the y-axis a boundary the graph cannot cross because division by zero is undefined.
- Horizontal Asymptote: Determined by examining the degrees of the polynomials in the numerator and denominator. Here, the horizontal asymptote is the x-axis itself, \( y = 0 \), which the graph approaches but never touches as \( x \) approaches infinity.
