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1. Definition of Rational Expressions

A rational expression is the ratio of two polynomials. It can be expressed in the form $R(x) = \frac{P(x)}{Q(x)}$, where:

• ( P(x) ) and ( Q(x) ) are polynomials
• ( Q(x) \neq 0 )

2. Domain of Rational Expressions

The domain of a rational expression is the set of all real numbers for which the expression is defined. This means we need to find values of $x$ that do not make the denominator zero.

Steps to find the domain:

1. Set the denominator $Q(x)$ equal to zero.
2. Solve for $x$ to find the values that must be excluded from the domain.

Example:

For the expression $R(x) = \frac{x^2 - 1}{x - 3}$:

The domain is all real numbers except $x = 3$:

3. Intercepts of Rational Expressions

a. x-intercepts:
The x-intercepts occur where the numerator $P(x)$ is zero (provided that $Q(x)$ is not also zero at that point).

b. y-intercepts:
The y-intercept occurs where $x = 0$. Substitute $x = 0$ into the expression $R(x)$.

Example:

For $R(x) = \frac{x^2 - 1}{x - 3}$:

• x-intercepts:
  Set the numerator to zero:

  $$x^2 - 1 = 0 \Rightarrow (x - 1)(x + 1) = 0$$

  Thus, $x = 1$ and $x = -1$.

• y-intercept:
  Substitute $x = 0$:

  $$R(0) = \frac{0^2 - 1}{0 - 3} = \frac{-1}{-3} = \frac{1}{3}$$

  So the y-intercept is $\left(0, \frac{1}{3}\right)$.

4. Asymptotes

a. Vertical Asymptotes:
Vertical asymptotes occur where the denominator is zero (and the numerator is not zero). These are found by setting $Q(x) = 0$.

b. Horizontal Asymptotes:
Horizontal asymptotes are determined by the degrees of the polynomials $P(x)$ and $Q(x)$.

• If the degree of $P(x) < \text{degree of } Q(x)$, then $y = 0$ is the horizontal asymptote.
• If the degree of $P(x) = \text{degree of } Q(x)$, then $y = \frac{a}{b}$ (where $a$ and $b$ are the leading coefficients of $P(x)$ and $Q(x)$, respectively) is the horizontal asymptote.
• If the degree of $P(x) > \text{degree of } Q(x)$, then there is no horizontal asymptote (but there may be an oblique asymptote).

Example:

For $R(x) = \frac{x^2 - 1}{x - 3}$:

• Vertical asymptote:
  From the earlier calculation, we found $Q(x) = 0$ at $x = 3$.

• Horizontal asymptote:
  Degree of $P(x) = 2$, degree of $Q(x) = 1$ (since $2 > 1$), so there is no horizontal asymptote.

5. Graphing Rational Expressions

To graph a rational expression, follow these steps:

1. Find the domain.
2. Determine intercepts.
3. Identify vertical and horizontal asymptotes.
4. Analyze the end behavior as $x$ approaches the asymptotes.
5. Plot additional points if needed.
6. Draw the graph, considering the asymptotes and intercepts.

Example: Graphing $R(x) = \frac{x^2 - 1}{x - 3}$

1. Domain: $(-\infty, 3) \cup (3, \infty)$
2. x-intercepts: $(1, 0)$ and $(-1, 0)$
3. y-intercept: $\left(0, \frac{1}{3}\right)$
4. Vertical asymptote: $x = 3$
5. Horizontal asymptote: None (since the degree of the numerator is higher than that of the denominator).

6. Additional Examples

Example 1: $R(x) = \frac{x + 2}{x^2 - 4}$

• Domain: Set $x^2 - 4 = 0$

  $$x^2 = 4 \Rightarrow x = 2, -2$$

  Domain:

  $$(-\infty, -2) \cup (-2, 2) \cup (2, \infty)$$

• Intercepts:

  • x-intercept: $x + 2 = 0 \Rightarrow x = -2$ (but it’s also excluded in the domain).
  • y-intercept: $R(0) = \frac{2}{-4} = -\frac{1}{2}$.

• Vertical asymptotes: $x = 2$ and $x = -2$ (both are excluded from the graph).

• Horizontal asymptote: Degree of $P = 1$ and Degree of $Q = 2 \Rightarrow y = 0$.

Example 2: $R(x) = \frac{2x^2 - 3x}{x^2 + x - 6}$

• Domain: Set $x^2 + x - 6 = 0$
  Factor:

  $$(x - 2)(x + 3) = 0 \Rightarrow x = 2, -3$$

  Domain:

  $$(-\infty, -3) \cup (-3, 2) \cup (2, \infty)$$

• Intercepts:

  • x-intercepts: Set $2x^2 - 3x = 0 \Rightarrow x(2x - 3) = 0 \Rightarrow x = 0$ or $x = \frac{3}{2}$.
  • y-intercept: $R(0) = 0$.

• Vertical asymptotes: $x = -3$ and $x = 2$.

• Horizontal asymptote: Degrees are equal, leading coefficients are $2$ and $1 \Rightarrow y = 2$.

Summary

• Rational expressions are ratios of polynomials.
• Domain is found by excluding values that make the denominator zero.
• Intercepts are found by setting the numerator to zero for x-intercepts and substituting $x = 0$ for the y-intercept.
• Asymptotes help understand the behavior of the graph as it approaches certain values.
• Graphing involves plotting these features to visualize the expression.

Feel free to ask if you need more examples or further clarification!