Problem 58 - Algebra 2

Question

Explain how to find the asymptotes of \(y=-\frac{3}{x-2}+11\)

Step-by-Step Solution

Answer

The asymptotes of the function \(y=-\frac{3}{x-2}+11\) are vertical asymptote at \(x=2\) and horizontal asymptote at \(y=11\).

Step 1: Determine Vertical Asymptote

The vertical asymptote can be found by setting the denominator equal to zero and solving for x, since the function is undefined at those points. So,\(x-2 = 0\)Solving for x gives: \(x = 2\)This means that the vertical asymptote is \(x=2\).

Step 2: Determine Horizontal Asymptote

The horizontal asymptote can be determined by considering the behavior of the function as x approaches positive or negative infinity.In this case, as x gets very large or very small, the \(\frac{3}{x-2}\) part of the function will get closer and closer to zero, and the function will tend towards 11. Thus, the horizontal asymptote is \(y=11\).

Key Concepts

Vertical Asymptote

Vertical asymptotes are important features of rational functions. They are the invisible lines that the graph of the function will approach but never touch or cross. These occur at values of the variable, typically denoted as \(x\), for which the function becomes undefined. To find vertical asymptotes, we generally focus on the denominator of a rational expression.

In our example, the function given is \(y = -\frac{3}{x-2} + 11\).

• The vertical asymptote can be located by setting the denominator \((x-2)\) equal to zero.
• This is done because rational functions are undefined where their denominators are zero.
• By solving \(x - 2 = 0\), we find \(x = 2\).
• This means the function will have a vertical asymptote at \(x = 2\).

Approaching this \(x\)-value from either side will cause the function’s value to increase or decrease without bound. It's crucial to remember that the function does not actually have a value at \(x = 2\) because it's undefined there.

Horizontal Asymptote

A horizontal asymptote of a rational function gives us a potential limit that the function's value can approach but not exceed, as \(x\) becomes very large in the positive or negative direction.

• For rational functions where the degree (the highest power of \(x\)) is larger in the numerator than in the denominator, the horizontal asymptote is \(y=0\).
• When the degrees are the same, the horizontal asymptote is the ratio of the leading coefficients.
• If the degree in the denominator is greater than that in the numerator, the horizontal asymptote is at \(y=0\).

In our function \(y = -\frac{3}{x-2} + 11\), there is an offset by \(+11\) after the division. As \(x\) gets very large or very small:

• The term \(-\frac{3}{x-2}\) becomes close to zero, as \(x\) increases or decreases greatly.
• Thus, \(y\) approaches \(11\).

Therefore, the horizontal asymptote is \(y = 11\). This indicates that no matter how large or small \(x\) becomes, the function's value won't exceed or drop below \(11\) at infinity.

Undefined Points

Undefined points in functions refer to values of \(x\) where the computation of the function becomes impossible, often due to division by zero. In the realm of rational functions, these points are closely related to the locations where the vertical asymptotes occur. Here is how undefined points in our example function are handled:

• The denominator of our rational part, \(x-2\), creates a situation where if \(x = 2\), the equation becomes problematic, as we cannot divide by zero.
• Thus, \(x=2\) is where the function becomes undefined.
• The function \(y = -\frac{3}{x-2} + 11\) has a discontinuity at this point.

These undefined points are crucial when sketching the graph of the function because they indicate breaks or holes that should not be connected by the curve. The concept reinforces that at \(x = 2\), the function doesn't exist, owing specifically to the restriction that dividing by zero is undefined in mathematics.