For the graph of y=f(x) a. Identify the x -intercepts. b....

For the graph of $y=f(x)$
a. Identify the $x$ -intercepts.
b. Identify any vertical asymptotes.
c. Identify the horizontal asymptote or slant asymptote if applicable.
d. Identify the $y$ -intercept.
$$
f(x)=\frac{(5 x-1)(x+3)}{x+2}
$$

Key Concepts

y-intercepts

The y-intercept is the point where the graph of the function crosses the y-axis, which is obtained by evaluating the function at x = 0, assuming the function is defined at that point. This point provides a useful reference for graphing and understanding the overall behavior of the function.

Horizontal and Slant Asymptotes

Horizontal asymptotes describe the behavior of a rational function as x approaches infinity or negative infinity and are determined by comparing the degrees of the numerator and denominator. When the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients; when the degree of the numerator is less, the horizontal asymptote is the x-axis; and when the numerator's degree is exactly one more than the denominator's, a slant (oblique) asymptote is present, which can be found by performing polynomial division.

Rational Function Analysis

Rational functions are quotients of polynomials and their graphs can exhibit features such as intercepts and asymptotes. The key aspects include determining where the function crosses the axes and understanding the behavior of the function near undefined points or at infinity. Analyzing these features provides insights into the overall structure and behavior of the function's graph.

x-intercepts

x-intercepts are the points where the graph of a function crosses the x-axis, which occur when the value of the function is zero. In the context of rational functions, these are determined by finding the values of x that make the numerator equal to zero, provided that these values do not also make the denominator zero, to ensure that the function is defined at those points.

Vertical Asymptotes

Vertical asymptotes are vertical lines where the function is undefined due to the denominator of the rational function equaling zero, and they indicate that the values of the function approach infinity or negative infinity near these lines. They are found by setting the denominator equal to zero and checking that the factor does not cancel with any factor in the numerator.

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