Problem 52 - Algebra 2

Question

Find the asymptotes of the graph of each equation. $$ y=\frac{2}{x+3} $$

Step-by-Step Solution

Answer

The vertical asymptote of the graph of the equation \(y = \frac{2}{x+3}\) is \(x = -3\). The graph of the function does not have a horizontal asymptote.

Step 1: Identify the Vertical Asymptote

The equation given is \(y = \frac{2}{x+3}\). In this equation, the denominator cannot be zero because division by zero is undefined. So solve \(x + 3 = 0\) to find the vertical asymptote. The solution \(x = -3\) will be the vertical asymptote

Step 2: Identify the Horizontal Asymptote

For rational functions, if the degree of the numerator is less than the degree of the denominator, then the horizontal asymptote is \(y = 0\). The equation here \(y = \frac{2}{x+3}\) has degree 0 in the numerator (2 is a constant, so it can be viewed as \(2x^0\)) and degree 1 in the denominator. Therefore, there is no horizontal asymptote because there is no number \(y\) that \(\frac{2}{x+3}\) approaches as \(x\) goes to plus or minus infinity.

Key Concepts

Vertical Asymptote

Finding a vertical asymptote is like searching for a point where a rational function wants to divide by zero. When the value in the denominator reaches zero, the function shoots up to infinity or plunges down to negative infinity, creating what we call a vertical asymptote. This is an invisible line on the graph – it acts like a wall that the function can never cross.

In the given equation, \[y = \frac{2}{x+3}, \]look at the denominator, which is \(x+3\). To find where it equals zero, solve for \(x\):

• Set the denominator equal to zero: \(x + 3 = 0\)
• Solve the equation: \(x = -3\)

Therefore, the vertical asymptote is at \(x = -3\).
Vertical asymptotes show how the graph behaves near undefined points. As you get closer to \(x = -3\), the graph will shoot up toward infinity or down toward negative infinity.

Horizontal Asymptote

Horizontal asymptotes help describe how a graph behaves as it stretches out to the left or right, toward negative or positive infinity. In the context of rational functions, these asymptotes show whether a function levels out to a constant value far from the origin.

Horizontal asymptotes depend on the degrees of the polynomial in both the numerator and the denominator. The degree of a polynomial is the largest exponent of the variable. For example, the degree of a constant, like 2, is zero, since it's technically \(2x^0\).

Let's apply this to the equation:\[y = \frac{2}{x+3}.\]

• The numerator (2) has degree 0
• The denominator \((x+3)\) has degree 1

If the degree of the numerator is less than the degree of the denominator, the function approaches zero as \(x\) grows larger.
Therefore, the horizontal asymptote here is \(y = 0\). This means that, far out on the graph, the curve will flatten out along the x-axis but never actually touch it.

Rational Functions

Rational functions are fractions where both the numerator and the denominator are polynomials. These functions are particularly interesting because of how they behave around asymptotes.
Analyzing a rational function gives insight into its structure and graph behavior, which can include:

• Vertical asymptotes, occurring when the denominator is zero and the function is undefined.
• Horizontal asymptotes, indicating the end behavior of the function as \(x\) goes to infinity.

In our example: \[y = \frac{2}{x+3}, \]The numerator (2) is a constant polynomial, and the denominator \((x+3)\) is a linear polynomial.
Understanding the interplay between these two components through asymptotes helps us sketch the graph and predict its behavior. Rational functions can model real-world phenomena, showing trends and patterns that involve division of variables.