Rational functions are an important part of Algebra I, especially for 11th-grade students. Knowing the main features of these functions helps in understanding more complicated math topics later on.

So, what is a rational function? It’s a type of function that can be written as $\frac{P(x)}{Q(x)}$. Here, $P(x)$ and $Q(x)$ are polynomials, which are just mathematical expressions that involve variables raised to different powers.

One key thing to know about rational functions is their domain. The domain includes all real numbers except where $Q(x)$ equals zero. Finding these special points is important because we want to avoid calculations that don't make sense.

Next, let’s talk about vertical asymptotes. A vertical asymptote happens at a point $x = a$ if $Q(a) = 0$ but $P(a)$ does not equal zero. This means that as $x$ gets closer to $a$, the function goes up to infinity or down to negative infinity. This behavior is unique to rational functions.

Now, let’s look at horizontal asymptotes. These help us understand what happens to the function as $x$ becomes really big or really small (positive or negative infinity). We can figure out the horizontal asymptotes based on the degrees (highest power) of the polynomials:

1. If the degree of $P(x)$ is less than that of $Q(x)$, the horizontal asymptote is $y = 0$.

2. If the degrees are the same, the asymptote is $y = \frac{a}{b}$, where $a$ and $b$ are the leading numbers from $P(x)$ and $Q(x)$.

3. If the degree of $P(x)$ is greater than that of $Q(x)$, there's no horizontal asymptote, but there might be a slant (or oblique) asymptote.

Intercepts are also very important. The y-intercept is where the function crosses the y-axis when $x = 0$. You can find it by plugging in 0 into the function: $\frac{P(0)}{Q(0)}$. The x-intercepts occur when $P(x) = 0$, which shows the points where the function crosses the x-axis.

Finally, it’s good to understand the continuity of rational functions. These functions are continuous everywhere except at their vertical asymptotes, where there is a break.

In conclusion, the key things to remember about rational functions are their definitions, what their domains are, vertical and horizontal asymptotes, intercepts, and their continuity. Knowing these details helps students work with rational expressions and equations better.