Rational function

Definition

A rational function is a type of function that can be expressed as the quotient of two polynomials, where the denominator is not zero. This function can exhibit a variety of behaviors depending on the degrees of the polynomials and the values of their coefficients. Rational functions are important because they often have asymptotes, intercepts, and specific intervals of increase or decrease that can be visualized through their graphs.

5 Must Know Facts For Your Next Test

1. Rational functions can have vertical asymptotes where the denominator equals zero, leading to undefined values.
2. Horizontal asymptotes in rational functions indicate the behavior of the function as the input approaches infinity, determined by the degrees of the numerator and denominator.
3. The graph of a rational function may cross its horizontal asymptote if the degree of the numerator is greater than or equal to the degree of the denominator.
4. Rational functions can have intercepts that are found by setting the numerator equal to zero and solving for x.
5. Discontinuities in rational functions occur at values where the denominator is zero, creating holes or vertical asymptotes in their graphs.

Review Questions

• How do vertical and horizontal asymptotes influence the graph of a rational function?
  • Vertical asymptotes occur at values where the denominator of a rational function is zero, causing the function to become undefined and leading to infinite behavior on either side of that value. Horizontal asymptotes describe the end behavior of the function as x approaches infinity or negative infinity. The relationship between the degrees of the numerator and denominator determines whether a horizontal asymptote exists, influencing how the graph behaves far from the origin.
• Discuss how finding intercepts and discontinuities helps in sketching the graph of a rational function.
  • Finding intercepts involves setting the numerator equal to zero to identify where the graph crosses the x-axis, while evaluating the function at zero gives the y-intercept. Discontinuities occur at points where the denominator equals zero, indicating potential holes or vertical asymptotes. These elements are crucial in sketching a rational function's graph because they provide key reference points that dictate how the graph behaves around those critical areas.
• Evaluate how changes in the coefficients of a rational function's polynomials affect its graph's features such as asymptotes and intercepts.
  • Changes in coefficients can significantly impact a rational function's graph by altering its steepness, position, and intercepts. For example, increasing a coefficient in the numerator raises all output values for that part of the function, potentially shifting intercepts vertically. Modifying coefficients in the denominator affects vertical asymptotes; if these values lead to new roots or eliminate existing ones, this can change how close or far away parts of the graph are from certain lines. Understanding these effects enables deeper insights into how polynomial behaviors shape overall rational function characteristics.

Related terms

Polynomial: A mathematical expression that consists of variables raised to non-negative integer powers and their coefficients, combined using addition, subtraction, and multiplication.

Asymptote:

A line that a graph approaches as it heads towards infinity, which can be horizontal, vertical, or oblique.

Discontinuity:

A point at which a function is not continuous, which can occur in rational functions due to division by zero.