5.4.5

Introduction

• This tutorial focuses on learning how to write possible rational equations from given x-intercepts.

Basic Concepts of Rational Functions

• A rational function is typically of the form:
  $y = \frac{f(x)}{g(x)}$.
• The x-intercepts of a function occur when the numerator is equal to zero.   - X-intercepts are identified in the numerator; each x-intercept corresponds to a factor in the numerator of the rational function.

Key Points on X-Intercepts

• When constructing the equation of a rational function:
    - Locate x-intercepts.   - Each x-intercept is accounted for by adding a factor to the numerator.

Examples of Rational Functions with X-Intercepts

Basic Example

• Start with the basic rational function:
  $y = \frac{1}{x}$.   - Observation: This function has no x-intercepts.

Adding an X-Intercept

• Add a factor to the numerator:
  $y = \frac{x-1}{x}$.   - Result: This function has an x-intercept at
  $x = 1$.

X-Intercept Limitations

• There are no limits to the number of x-intercepts that a rational function can possess.

Example with Multiple X-Intercepts

• A function might have several x-intercepts; consider a function with five x-intercepts:
    - Observations indicate x-intercepts at:
      - $x = -3$
      - $x = 1$
      - $x = 2$
      - $x = 3$
      - $x = 5$
• Each factor in the numerator correlates directly with an x-intercept on the graph.

Practice Problems

Example Problem 1

• Given x-intercepts at:
    - $x = -2$
    - $x = 4$
    - $x = 5$
• Procedure to write a possible rational equation:
    1. Start with a basic rational function, leaving the numerator blank:
  $y = \frac{?}{g(x)}$.   2. Add factors corresponding to each x-intercept:
       - First Factor: $x + 2$ for the intercept at $x = -2$.
       - Second Factor: $x - 4$ for the intercept at $x = 4$.
       - Third Factor: $x - 5$ for the intercept at $x = 5$.   3. Possible equation:
  $y = \frac{(x + 2)(x - 4)(x - 5)}{g(x)}$.

Example Problem 2

• Given x-intercepts at:
    - $x = -1$
    - $x = 1$
• Procedure to write a possible rational equation:
    1. Start with a basic rational function, leaving the numerator blank:
  $y = \frac{?}{g(x)}$.   2. Add factors corresponding to each x-intercept:
       - First Factor: $x + 1$ for the intercept at $x = -1$.
       - Second Factor: $x - 1$ for the intercept at $x = 1$.
    3. Possible equation:
  $y = \frac{(x + 1)(x - 1)}{g(x)}$.

Conclusion

• In this lesson, you learned how to write a rational equation based on the location of x-intercepts:   - Each x-intercept is represented as a factor in the numerator, articulating its location in the equation of the rational function.