Problem 56 - Algebra 2

Question

Find the asymptotes of the graph of each equation. $$ y=\frac{-2}{5-x}-6 $$

Step-by-Step Solution

Answer

The vertical asymptote of the graph is at \(x = 5\), and the horizontal asymptote is at \(y = -6\).

Step 1: Rewrite the Function

To make it easier to identify x-values at which the function is undefined, rewrite the function \(y=\frac{-2}{5-x}-6\) to the equivalent form \(y=\frac{-2}{x-5}-6\).

Step 2: Identify Vertical Asymptotes

A vertical asymptote occurs at the x-values where the denominator of the function equals zero. To find the vertical asymptote, set x-5 equal to zero and solve for x. In this case, if you set \(x - 5 = 0\), it leads to \(x = 5\). So, the vertical asymptote is \(x = 5\).

Step 3: Identify Horizontal Asymptotes

A horizontal asymptote is determined by comparing the degrees of the numerator and the denominator. In this case, the degree of the numerator is 0 and the degree of the denominator is 1. When the degree of the denominator is larger, the x-axis \(y = 0\) is the horizontal asymptote. However, in this case, there is a constant of \(-6\) added to the function, shifting it down by 6 units. So, the horizontal asymptote will be at \(y = -6\).

Key Concepts

Vertical Asymptotes

Vertical asymptotes in a graph indicate where a function is undefined. These occur when the denominator of a rational function is zero. Solving the equation where the denominator equals zero helps pinpoint these vertical lines.

• For the function given, \( y = \frac{-2}{5-x} - 6 \), rewrite it as \( y = \frac{-2}{x-5} - 6 \) for clarity.
• Now set the denominator, \( x-5 \), equal to zero: \( x - 5 = 0 \).
• This simplifies to \( x = 5 \), establishing that the vertical asymptote is at \( x = 5 \).

Vertical asymptotes represent locations along the x-axis where the function shoots up to infinity or down to negative infinity as it approaches those x-values.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as it heads toward infinity either in the positive or negative direction on the x-axis. These lines suggest how a function behaves further away from the origin.

• To determine horizontal asymptotes, compare the degrees of the numerator and the denominator.
• In \( y = \frac{-2}{x-5} - 6 \), the numerator has a degree of 0 (a constant), while the denominator has a degree of 1.
• When the degree of the denominator is greater than the numerator, the horizontal asymptote is usually at \( y = 0 \).
• However, since there is a \(-6\) in the function, it adjusts the horizontal asymptote to \( y = -6 \).

This adjustment implies a shift of the horizontal asymptote downwards by 6 units, showing that the function levels off at this line as x grows larger or smaller.

Rational Functions

Rational functions come from dividing one polynomial by another. Understanding this helps in analyzing asymptotic behavior.

• The given function \( y = \frac{-2}{x-5} - 6 \) is an example of a rational function.
• The asymptotes in rational functions provide crucial insights into their graphs, indicating undefined points and long-term trends.
• Vertical asymptotes occur where the denominator equals zero, showing sharp disruptions in the graph.
• Horizontal asymptotes display the behavior of the function as x goes to infinity.

By understanding these elements, you can graph rational functions more precisely, highlighting their asymptotic direction and areas where they are not defined.