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

They are the x-intercepts of the graph, where the function's value is zero, provided they are in the domain.

They indicate where the function is undefined, leading to vertical asymptotes or holes in the graph.

Zeros and vertical asymptotes are the points where a rational function can change its sign (positive to negative or vice versa).

It involves testing values within each interval created by zeros and asymptotes to determine the function's sign in that interval.

To ensure that the zeros found are actually part of the function and not points where the function is undefined.

The function approaches infinity (or negative infinity) as x gets closer to the asymptote.

Simplification helps to identify and remove any removable discontinuities (holes) and avoids incorrect zero identification.

By using interval analysis, testing values in each interval defined by zeros and asymptotes.

Factoring helps to identify zeros and simplify the function, making it easier to analyze its behavior.

Rational functions are built from polynomial functions, and understanding polynomial zeros and behavior is essential for analyzing rational functions.