$r(x) = \frac{P(x)}{Q(x)}$, where P(x) and Q(x) are polynomials.

Solve $P(x) = 0$, ensuring that the solutions are not zeros of $Q(x)$.

$(-\infty, a)$, $(a, b)$, $(b, \infty)$ where a and b are critical points.

$Q(a) = 0$ and $P(a) \ne 0$ for $r(x) = \frac{P(x)}{Q(x)}$.

Analyze the sign of $r(x)$ in intervals defined by zeros and vertical asymptotes.

$ax^2 + bx + c = a(x - r_1)(x - r_2)$, where $r_1$ and $r_2$ are the roots.

Domain = {$x \in \mathbb{R} | Q(x) \ne 0$}

$\frac{P(x)}{Q(x)} = \frac{(x-a)R(x)}{(x-a)S(x)} = \frac{R(x)}{S(x)}$, if $x \ne a$.

$y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

$r(x) = \frac{(x-a)P(x)}{(x-a)}$, where the discontinuity is at $x=a$.

A function that can be expressed as the quotient of two polynomials.

The real zeros of the numerator that are also in the domain of the function.

A vertical line x = a where the function approaches infinity or negative infinity as x approaches a.

All real numbers except for the values that make the denominator equal to zero.

The process of determining the sign of a rational function on different intervals defined by its zeros and asymptotes.

The zeros of the numerator and the zeros of the denominator.

The x-values where the function equals zero (numerator zeros).

The x-values where the function is undefined (denominator zeros).

Values used to check the sign of the function in each interval created by zeros and asymptotes.

To cancel out common factors in the numerator and denominator.

It represents an x-intercept, where the function's value is zero.

It represents a point where the function is undefined and approaches infinity or negative infinity.

These are the intervals where the graph is above the x-axis.

These are the intervals where the graph is below the x-axis.

It represents a removable discontinuity, where a factor in the numerator and denominator cancels out.

The graph approaches infinity or negative infinity as $x$ approaches the asymptote.

It means $x=a$ is a real zero of the rational function.

The domain consists of all x-values except those at vertical asymptotes or holes.

It describes how the function behaves as x approaches positive or negative infinity.

If the graph approaches a constant y-value as x goes to positive or negative infinity.