Rational functions are an interesting part of Algebra II, especially for 11th graders. So, what exactly are they?

A rational function is simply a fraction where both the top and bottom are polynomials. We write it like this:

$$f(x) = \frac{P(x)}{Q(x)}$$

Here, $P(x)$ is the polynomial on top, and $Q(x)$ is the polynomial on the bottom. It's important that $Q(x)$ is not zero. If it is zero, the function can't be defined, which can change how the graph looks.

How to Identify Rational Functions

Identifying a rational function is easy if you follow these steps:

1. Look at the Form: Check if the function is a fraction made from two polynomials. For example, $f(x) = \frac{2x^2 + 3}{x - 1}$ is a rational function.

2. Check Degrees: Pay attention to the degrees of the polynomials. The degree is the highest power of $x$ in the polynomial. If the degree of the top (numerator) is just one more than the bottom (denominator), the function will go towards positive or negative infinity when $x$ gets close to certain numbers.

Important Features to Look At

When you graph rational functions, there are some key features to consider:

1. Vertical Asymptotes: These happen where the bottom part (denominator) equals zero but the top part (numerator) does not. For example, in $f(x) = \frac{2x^2 + 3}{x - 1}$, setting $x - 1 = 0$ gives us $x = 1$ as a vertical asymptote.

2. Horizontal Asymptotes: These show how the function behaves as $x$ gets really big or really small. Generally:

   • If the degree of the top is less than the bottom, the horizontal asymptote is $y = 0$.
   • If the degrees are the same, the horizontal asymptote equals the ratio of the leading coefficients.

3. Intercepts: To find the $y$-intercept, just plug in $x = 0$ into the function. For the $x$-intercepts, set the top part (numerator) equal to zero and solve for $x$.

How to Graph Rational Functions

Here’s a simple step-by-step approach to graph a rational function:

1. Find Asymptotes: Start by figuring out the vertical and horizontal asymptotes. Draw dashed lines for these on your graph.

2. Find Intercepts: Calculate and plot the $x$ and $y$ intercepts. These are important points on your graph.

3. Examine Behavior Around Asymptotes: Look at how the function behaves as it gets close to the asymptotes from both sides. This helps you understand the graph better.

4. Plot More Points: Pick a few more $x$ values, calculate $f(x)$ for them, and plot these points.

5. Connect the Dots: Finally, draw a smooth curve connecting all the points, taking into account the asymptotes and how the function behaves.

Example

Let’s look at an example of a rational function:

$$f(x) = \frac{x^2 - 1}{x^2 - 4}$$

• Vertical Asymptotes: Set $x^2 - 4 = 0$. This gives $x = 2$ and $x = -2$.
• Horizontal Asymptote: Since both polynomials have the same degree, the horizontal asymptote is $y = \frac{1}{1} = 1$.
• Intercepts: The numerator gives $x$-intercepts at $x = 1$ and $x = -1$. Also, when you find $f(0)$, it equals $-\frac{1}{4}$, which means the $y$-intercept is $-\frac{1}{4}$.

By following these steps, you can make identifying and graphing rational functions a fun and easy activity! Happy graphing!