Make a hand-drawn graph. Be sure to label all the...

Make a hand-drawn graph. Be sure to label all the asymptotes. List the domain and the $x$ -intercepts and the $y$ -intercepts. Check your work using $a$ graphing calculator.
$$f(x)=\frac{1}{x-5}$$

Key Concepts

Intercepts

Intercepts are the points where the graph crosses the axes. The x-intercepts are found by setting the function equal to zero and solving for x, while the y-intercept is found by evaluating the function at x = 0. These points provide key anchors for constructing an accurate graph.

Graphing Rational Functions

This concept involves sketching the behavior of a function that is the ratio of two polynomials. It requires identifying key features such as asymptotes, intercepts, and the overall shape of the graph, as well as understanding how the function behaves near critical points where it may be undefined or approach infinity.

Domain of a Function

The domain is the set of all x-values for which the function is defined. For rational functions, this typically means excluding any value that causes the denominator to be zero, as division by zero is undefined.

Vertical Asymptotes

Vertical asymptotes are vertical lines that the graph of a function approaches but never touches. They occur at the x-values that make the denominator zero (and are not canceled out by factors in the numerator), indicating that the function grows without bound as it approaches these values.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of rational functions as x approaches positive or negative infinity. They are determined by comparing the degrees of the numerator and the denominator, and they indicate the value that the function approaches for very large or very small x.

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