Problems 70-79 are based on material learned earlier in the...

Problems $70-79$ are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Graph: $R(x)=\frac{2 x+1}{x-3}$

Key Concepts

Intercepts

Intercepts, including x-intercepts and y-intercepts, are the points where the function crosses the respective axes. Finding these intercepts often involves setting the numerator to zero or evaluating the function at a specific x-value, helping to anchor the graph in the coordinate plane.

Domain Restrictions

The domain of a rational function includes all real numbers except those that make the denominator equal to zero. Identifying these restrictions is crucial for understanding where the function exists and how it behaves.

Graph Transformations

Graph transformations involve modifying a basic function through translations, stretches, or reflections to create a new graph. With rational functions, these transformations help in understanding shifts in asymptotes and intercepts relative to the parent function.

Rational Functions

Rational functions are expressions that comprise the quotient of two polynomials. They are characterized by their potential discontinuities and asymptotic behavior and are an essential part of algebra and pre-calculus studies.

Vertical and Horizontal Asymptotes

Vertical asymptotes occur in rational functions where the denominator is zero, indicating values at which the function is undefined. Horizontal asymptotes describe the behavior of the graph as the input becomes very large or very small, typically determined by the degrees of the numerator and denominator.

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