In Exercises 16-20 , match the correct graph A-E to the...

In Exercises $16-20$ , match the correct graph $A-E$ to the function without using your calculator. Then, after you have answered all of them, if you have a graphing calculator, use your calculator to check your answers. Each graph in this group is plotted on $[-6,6]$ by $[-6,6] .$ Hint: Consider the asymptotes. (If you try graphing these graphs in Connected mode rather than Dot mode, you will see some lines that are not part of the graph, but the result of the calculator connecting disconnected parts of the graph.) GRAPHS CANNOT COPY $$y=\frac{2 x^{2}+3}{x^{2}-1}$$

In Exercises $16-20$ , match the correct graph $A-E$ to the function without using your calculator. Then, after you have answered all of them, if you have a graphing calculator, use
your calculator to check your answers. Each graph in this group is plotted on $[-6,6]$ by $[-6,6] .$ Hint: Consider the asymptotes. (If you try graphing these graphs in Connected
mode rather than Dot mode, you will see some lines that are not part of the graph, but the result of the calculator connecting disconnected parts of the graph.)
GRAPHS CANNOT COPY

$$y=\frac{2 x^{2}+3}{x^{2}-1}$$

Key Concepts

Graphing Rational Functions Without a Calculator

Graphing rational functions manually involves identifying key features such as vertical and horizontal asymptotes, holes (if any), and overall end behavior. By analyzing these characteristics, one can sketch an approximate graph by plotting strategic points and understanding how the function behaves near its discontinuities and at infinity. This approach is essential for validating the overall shape of the graph before using any technological tools for verification.

Rational Functions

Rational functions are expressions defined as the quotient of two polynomials. They are important in understanding the behavior of complex functions because their graphs can exhibit various features such as asymptotes, discontinuities, and distinct end behaviors, which arise from the polynomial terms in the numerator and denominator.

Vertical Asymptotes

Vertical asymptotes occur in rational functions at the values of the independent variable where the denominator equals zero and the numerator does not also vanish in such a way that the discontinuity is removable. These lines represent values that the function will approach but never actually reach, and they are key in understanding the function’s behavior near points of discontinuity.

Horizontal Asymptotes

Horizontal asymptotes describe the end behavior of a rational function as the independent variable approaches positive or negative infinity. When the degrees of the numerator and denominator are equal, the horizontal asymptote is determined by the ratio of the leading coefficients, providing insight into the long-term behavior of the function.

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