Problem 21 - Algebra 2

Question

Sketch the asymptotes and the graph of each equation. \(y=\frac{-8}{x+5}-6\)

Step-by-Step Solution

Answer

The graph of the given rational function will have vertical asymptote at \(x=-5\) and horizontal asymptote at \(y=-6\). The graph of the equation will approach these lines but does not touch them.

Step 1: Identifying the Vertical Asymptote

Find the vertical asymptote by setting the denominator equal to zero and solve it: \(x+5=0\), so \(x=-5\) is a vertical asymptote.

Step 2: Identifying the Horizontal Asymptote

If the degree of the denominator (x) is greater than the degree of numerator (constant term), the horizontal asymptote will be at \(y=c\), where c is the constant. So in this case, our horizontal asymptote will be \(y=-6\).

Step 3: Sketching the Graph

Using asymptotes, plot them on the graph. The plot will not cross a vertical asymptote, and as the \(x-\)values get larger or smaller, the plot will get closer to the horizontal asymptote but never reach it, which is \(y=-6\). Draw a curve for the plot that fits these characteristics.

Key Concepts

Understanding Vertical Asymptotes

Vertical asymptotes occur where the denominator of a rational function equals zero, and the function approaches infinity. Imagine driving down a road that abruptly has a barrier blocking the way. This barrier is similar to a vertical asymptote on a graph where the line cannot bypass it and shoots upward or downward drastically instead.

In the original exercise, the function given is \(y=\frac{-8}{x+5}-6\). To find where the vertical asymptote is, we set the denominator equal to zero: \(x+5=0\). Upon solving, we find \(x=-5\). This is our vertical asymptote.

When sketching graphs, vertical asymptotes are represented by dashed vertical lines. The graph of the function will approach these lines but never actually touch or cross them. This helps indicate the function's behavior near these critical values.

Identifying Horizontal Asymptotes

Horizontal asymptotes provide information on the end behavior of a rational function. If vertical asymptotes are barriers in a road, horizontal asymptotes are guidelines that show where the road is headed at the edges of the graph.

The determination of a horizontal asymptote depends on the degrees (highest power of x) of the polynomial in the numerator versus the denominator. In the case of \(y=\frac{-8}{x+5}-6\), the numerator has a degree of zero (since it is a constant), and the denominator has a degree of one.

When the degree of the denominator is greater than that of the numerator, the horizontal asymptote is at \(y=c\), where \(c\) is the constant term outside the fraction. Thus, we conclude that the horizontal asymptote is at \(y=-6\). This means that as \(x\) values increase or decrease immensely, the function will get infinitely closer to this line, but not actually meet it.

Graphing Rational Functions

Graphing rational functions involves integrating both vertical and horizontal asymptotes to understand the entire picture. Let's walk through graphing the function \(y=\frac{-8}{x+5}-6\).

• Step 1: Plot the vertical asymptote at \(x = -5\) as a dashed line. This signals where the graph will shoot upwards or downwards and acts as a boundary.
• Step 2: Plot the horizontal asymptote at \(y = -6\) with another dashed line. It denotes the trend of the graph as \(x\) moves towards positive or negative infinity.
• Step 3: Draw the graph. Ensure the curves approach but do not cross these asymptotes. As \(x\) gets smaller, the function will decrease toward \(y = -6\) from the upper side and as \(x\) increases, it moves upward away from \(y = -6\).

Vertical asymptotes reflect rapid changes, while horizontal ones show long-term behavior. Together, they give a complete vision of how the rational function behaves.