How to graph a reciprocal function?
Understand the Problem
The question is asking how to create a graph of a reciprocal function, which typically has the form y = 1/x. This involves understanding the characteristics of reciprocal functions, such as their asymptotes and behavior as x approaches 0.
Answer
The graph of $y = \frac{1}{x}$ has two branches in the first and third quadrants, with vertical asymptote at $x = 0$ and horizontal asymptote at $y = 0$.
Answer for screen readers
The graph of the reciprocal function $y = \frac{1}{x}$ consists of two branches, located in the first and third quadrants, with vertical asymptote at $x = 0$ and horizontal asymptote at $y = 0$.
Steps to Solve
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Identify the function
The reciprocal function can be expressed as $y = \frac{1}{x}$. -
Determine asymptotes
Reciprocal functions have two types of asymptotes: the vertical asymptote and the horizontal asymptote. For $y = \frac{1}{x}$, there is a vertical asymptote at $x = 0$ and a horizontal asymptote at $y = 0$. -
Evaluate specific points
To sketch the graph, select several values of $x$ to find corresponding $y$ values:- For $x = 1$, $y = \frac{1}{1} = 1$.
- For $x = -1$, $y = \frac{1}{-1} = -1$.
- For $x = 2$, $y = \frac{1}{2} = 0.5$.
- For $x = -2$, $y = \frac{1}{-2} = -0.5$.
- For $x = 0.5$, $y = \frac{1}{0.5} = 2$.
- For $x = -0.5$, $y = \frac{1}{-0.5} = -2$.
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Sketch the graph
Using the points gathered and the asymptotes, plot the points on a coordinate plane. Draw the curves approaching the vertical asymptote at $x = 0$ and the horizontal asymptote at $y = 0$. You should see two branches of the graph: one in the first quadrant and one in the third quadrant. -
Analyze the graph behavior
As $x$ approaches 0 from the positive side, $y$ approaches infinity, and as $x$ approaches 0 from the negative side, $y$ approaches negative infinity. As $x$ goes to positive or negative infinity, $y$ approaches 0.
The graph of the reciprocal function $y = \frac{1}{x}$ consists of two branches, located in the first and third quadrants, with vertical asymptote at $x = 0$ and horizontal asymptote at $y = 0$.
More Information
The reciprocal function, $y = \frac{1}{x}$, is a classic example in mathematics and demonstrates concepts such as asymptotic behavior, which are fundamental in understanding limits and function graphs.
Tips
- Forgetting to identify asymptotes can lead to an inaccurate graph representation. Always include them in your graph.
- Mixing up the signs when evaluating reciprocal values at negative or small fractions can result in incorrect points on the graph.
