Rational functions refer to any ratio of two polynomials. Graphs of rational functions are a way to visually represent equations where the variables are directly, inversely, or jointly related to one another.

**Variation: **Equation that relates a variable to one or more other variables by multiplication and division.

**Direct Variation: **Related variables increase or decrease together at a steady rate.

**Inverse Variation: **When one variable increases, the other decreases, and vice versa.

**Joint Variation: **One variable varies as a product of two or more variables.

**Asymptote: **Values that a function approaches but never reaches.

General equations for different types of** variations:**

**Direct variation:**\(y = kx\)- Example: distance = speed × time (the distance something travels is equal to the speed it is moving at and how long it is moving)
**Inverse variation:**\(y = \frac{k}{x}\)**Joint variation:**\(z = kxy\)- \(z\) is the product of both \(x\) and \(y\)
- Example: \(V = \pi R^2 ∙ h\) (the volume of a cylinder equals the radius of the base squared times the height of the cylinder times the constant of proportionality \(π\))
- In all three variations, \(k\) is called the constant of proportionality and is not equal to \(0\).

An equation can show one, two, or all three types of variations. A equation with all three types of variation shows combined variation.

Graphing rational functions is not that different from graphing other types of equations, except we have to consider asymptotes.

Direct and inverse variations have very different looking graphs.

- Linear equations where \(b\) in the slope-intercept form equals \(0\) are direct variations (such that \(y = mx\)).
- Graphs of inverse variation equations are also called hyperbola.
- They have two branches that approach but never cross lines called
**asymptotes**. - A vertical asymptote is a vertical line that x will never intersect.
- A horizontal asymptote is a horizontal line that y will never intersect.

Vertical asymptotes (or x-asymptotes) are found at the values of \(x\) that would make the denominator equal to \(0\).

- Example: \(y = \frac{1}{(x - 2)^2}\)

If \(x = 2\), the denominator equals \(0. x = 2\) is the vertical asymptote. The graph also has a horizontal asymptote at \(y = 0\).

- A function does not have to have any vertical asymptotes.

Horizontal asymptote (or y-asymptote) is the value that \(y\) approaches for very large \(±x\). We need to consider several cases. Look at the highest power of \(x\) in the numerator and the denominator.

- The highest power in the numerator is less than the highest power in the denominator, the horizontal asymptote is \(y = 0\)
- If the highest power in the numerator is equal to the highest power in the denominator, the horizontal asymptote is \(y = \frac{\text{coefficient of highest power of x}}{\text{coefficient of highest power of} x^.}\)
- If the highest power in the numerator is greater than the highest power in the denominator, there is no horizontal asymptote. There could be an asymptote that is slanted or no asymptote at all.

Example: Consider \(y = \frac{2x^2 + x + 3}{3x^2 - 2x + 1}\). Plug in a large value of x, such as 100.

\(y = \frac{2(100)^2 + 100 + 3}{3(100)^2 - 2(100) + 1} = \frac{20000 + 100 + 3}{30000 - 200 + 1}\)

We can see that the first terms in the numerator and the denominator are much bigger than the other terms. The horizontal asymptote is

\(y = \frac{2x^2}{3x^2} = \frac{2}{3}\).

Example: Now consider \(y = \frac{x + 3}{3x^2 - 2x + 1}\). Plugin \(x = 100\).

\(y = \frac{100 + 3}{3(100)^2 - 2(100) + 1} = \frac{100 + 3}{30000 - 200 + 1}\)

The first term in the denominator is much bigger than the first term in the numerator. The horizontal asymptote is \(y = 0\).

Example: \(\frac{4x^2 + 3x + 2}{x - 1}\) can be rewritten as \(y = 4x + 7 + \frac{9}{x - 1}\) by doing long division. Plug in \(x = 100\).

\(\frac{9}{x - 1}\) approaches \(0\), leaving \(y = 4x + 7. y = 4x + 7\) is the slanted asymptote (oblique asymptote), so the function approaches this line for large values of x.

Graph \(y = \frac{-x^2}{x^2 - 4}\).

To find the vertical asymptote, set the denominator equal to zero: \(x^2-4 = 0\). We can factor to find that \(x = ±2\), so \(x = 2\)

and \(x = -2\) are the vertical asymptotes. To find the horizontal asymptote: \(y = \frac{-x^2}{x^2} = -1\).

Graph all of the asymptotes first (the red dotted lines). The rest is like graphing anyother equation. Create a table of values and plot the points. Make sure to includepoints near the vertical asymptotes!