$r(x) = \frac{P(x)}{Q(x)}$, 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 $x=a$.

Zeros: function equals zero | Vertical Asymptotes: function is undefined

Removable: factor cancels out, hole in graph | Non-removable: vertical asymptote

$r(x) = 0$: find zeros | $r(x) > 0$: interval analysis

Numerator: determines zeros | Denominator: determines asymptotes

Polynomial: all real numbers | Rational: excludes zeros of denominator

Endpoints: function is zero | Asymptotes: function is undefined

$r(x) > 0$: find intervals where function is positive | $r(x) < 0$: find intervals where function is negative

Near a zero: the function crosses or touches the x-axis | Near a vertical asymptote: the function approaches infinity or negative infinity

Graphically: identify x-intercepts | Algebraically: solve for x when the function equals zero

Rational equations: find specific values | Rational inequalities: find intervals

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.