Lesson 78: Graphing Rational Functions

New Concept

A rational function is a function whose rule can be given as a rational expression.

What’s next

Next, you’ll use this idea to find asymptotes—the invisible boundaries of the function—and graph these new curves.

Property

A rational function is a function whose rule can be given as a rational expression. This means that a rational function has a variable in the denominator. A value of a variable for which a function is undefined is called an excluded value.

Explanation

Think of a rational function as a fraction on a mission, but with one critical rule: never, ever divide by zero! An 'excluded value' is the one number for the variable that would break this sacred rule, causing the function to have a meltdown. It’s the single value of x that is officially banned from the math party.

Examples

For $y = \frac{x}{x-5}$, the excluded value is $x=5$, because the denominator would be $5-5=0$.
For $y = \frac{x+1}{2x+8}$, the excluded value is $x=-4$, because $2(-4)+8=0$.
For $y = \frac{10}{x}$, the excluded value is $x=0$.

Property

To find the excluded values for a rational function, set the expression in the denominator equal to zero and solve for the variable. The solution is the value that is excluded from the function's domain.

Explanation

Finding an excluded value is like being a detective hunting for the one number that causes mathematical chaos. Your mission is to isolate the denominator, set it equal to zero, and solve the mystery. This reveals the variable’s value that isn't allowed, keeping your function stable and preventing it from breaking down into total mathematical nonsense.

Examples

For $y = \frac{m-1}{5m-10}$: Set $5m-10=0$. This simplifies to $5m=10$, so the excluded value is $m=2$.
For $y = \frac{8}{x+3}$: Set $x+3=0$. This means the excluded value is $x=-3$.
For $y = \frac{2m}{3m}$: Set $3m=0$. The excluded value is $m=0$.

An expression is undefined when its denominator is zero; let's find that forbidden value. This example will focus on the key idea of determining excluded values.

Example Problem

Find the excluded value for $y = \frac{k-5}{4k-20}$.

Property

An asymptote is a boundary line that the graph of a function approaches but never touches or crosses. For a function in the form $y = \frac{a}{x-b} + c$, the vertical asymptote occurs at $x=b$ and the horizontal asymptote occurs at $y=c$.

Explanation

Imagine your graph is a super shy puppy and the asymptote is an invisible electric fence. The puppy can get really, really close to the fence, sniffing along the edge from either side, but it will never actually touch or cross it. These invisible 'fences' are essential guides that define the shape of your graph and its boundaries.

Examples

For the function $y = \frac{5}{x-3} + 2$, the vertical asymptote is $x=3$ and the horizontal asymptote is $y=2$.
For the function $y = \frac{-4}{x+6} - 1$, the vertical asymptote is $x=-6$ and the horizontal asymptote is $y=-1$.
For the function $y = \frac{1}{x}$, the vertical asymptote is $x=0$ and the horizontal asymptote is $y=0$.

Every rational function has invisible guide lines that shape its graph; let's find them. This example demonstrates the key idea of determining asymptotes from the function's equation.

Example Problem

Identify the asymptotes of the function $y = \frac{5}{x+7} - 2$.

Property

Step 1: Identify and graph the vertical ($x=b$) and horizontal ($y=c$) asymptotes with dashed lines. Step 2: Make a table of values by choosing x-values on both sides of the vertical asymptote. Step 3: Plot the points and connect them with smooth curves.

Explanation

Graphing a rational function is like an advanced connect-the-dots puzzle with invisible force fields. First, you draw your dashed-line asymptotes—the 'no-go' zones for your graph. Then, you plot a few key points on either side of the vertical line. Finally, connect the dots with smooth curves that get incredibly close to the asymptotes but never touch.

Examples

To graph $y = \frac{1}{x-2} + 1$: Asymptotes are $x=2$ and $y=1$. Plot points like $(1,0)$ and $(3,2)$ to draw the curves.
To graph $y = \frac{-2}{x+3}$: Asymptotes are $x=-3$ and $y=0$. Plot points like $(-4, 2)$ and $(-2, -2)$ to see the shape.