Question

The graph of \(y=\frac{3 x-6}{x-4}\) has a vertical asymptote. What is it?

Short answer

The vertical asymptote is at \( x = 4 \).

Step-by-step solution

Identify the Vertical Asymptote

To find the vertical asymptote of the rational function, set the denominator equal to zero. The function is given by \( y = \frac{3x - 6}{x - 4} \). We need to solve for \(x \text{when} \ (x - 4) = 0 \).

Solve for x

Set the denominator equal to zero and solve for \(x\). \( x - 4 = 0 \) then \( x = 4 \).

Interpret the Result

The value \( x = 4 \) is where the denominator becomes zero, and thus the graph of the function has a vertical asymptote at this point.

Key concepts

Rational Functions

Rational functions are a type of function expressed as the ratio of two polynomials. The general form is: \[ y = \frac{P(x)}{Q(x)} \] where \(P(x)\) and \(Q(x)\) are polynomials.
These functions can exhibit interesting behaviors, especially around points where the denominator is zero.
Key characteristics of rational functions include:

• Having possible vertical and horizontal asymptotes.
• Poles, where the function goes to infinity.
• Holes, where the function is undefined but does not reach infinity.

Understanding these concepts helps in effectively analyzing and graphing rational functions.

Denominator

The denominator of a rational function plays a crucial role in determining its behavior.
A zero denominator causes the function to become undefined at that point.
Key points to remember regarding denominators are:

• Setting the denominator equal to zero allows us to locate vertical asymptotes.
• Any value of \(x\) that makes the denominator zero will make the function undefined.
• The solutions to this equation are used to find important features like vertical asymptotes and points of discontinuity.

Using the example \( y = \frac{3x - 6}{x - 4} \), the denominator \( x - 4 \) equals zero when \(x = 4\).

Asymptotes

Asymptotes are lines that a graph approaches but never actually reaches.
There are two main types: vertical and horizontal. Vertical asymptotes occur where the denominator is zero, while horizontal asymptotes are determined by the degrees of the polynomials in the function.
Vertical asymptotes:

• Indicate points of discontinuity where the function tends to infinity.
• Found by setting the denominator equal to zero.
• Example: In \( y = \frac{3x - 6}{x - 4} \), the vertical asymptote is at \(x = 4\) because \(x - 4 = 0\).

Recognizing asymptotes helps predict the behavior of the function near points of discontinuity.