Problem 34 - Algebra and Trigonometry

Question

Find the vertical asymptotes, if any, and the values of \(x\) corresponding to holes, if any, of the graph of each crational function. $$ h(x)=\frac{x+6}{x^{2}+2 x-24} $$

Step-by-Step Solution

Answer

The function has a vertical asymptote at \(x=4\) and a hole at \(x=-6\)

Step 1: Define the function

The given function is \(h(x)=\frac{x+6}{x^{2}+2 x-24}\)

Step 2: Search for vertical asymptote

To find the vertical asymptote, set the denominator equal to zero and solve for \(x\). \[x^{2}+2x-24=0\] Using the factorization method, \[(x-4)(x+6)=0\] So the solutions are \(x=4\) and \(x=-6\)

Step 3: Identify holes in the graph

To determine if there are any holes in the graph, we observe if there are any values that makes both the numerator and the denominator equal to zero simultaneously. At \(x=-6\), both the numerator and the denominator are zero, so the function has a hole at \(x=-6\)

Step 4: Conclusion

The function \(h(x)=\frac{x+6}{x^{2}+2 x-24}\) has a vertical asymptote at \(x=4\) and a hole at \(x=-6\)

Key Concepts

Rational Functions

Rational functions are like fractions, but with polynomials. They have an expression in the numerator and another one in the denominator. An example is the function \( h(x)=\frac{x+6}{x^{2}+2x-24} \). The polynomial in the numerator is \( x+6 \) and the polynomial in the denominator is \( x^{2}+2x-24 \).

Understanding the role of each part is important. The numerator determines the zeros of the function, impacting where the function crosses the \( x \)-axis. The denominator impacts the function's discontinuities. These discontinuities often show up as vertical asymptotes or holes in graphs, both crucial for analyzing a rational function.

Remember, the key features of rational functions often include vertical and horizontal asymptotes and holes, which all tell us something important about the behavior of the graph.

Holes in Graphs

Holes in graphs occur when a rational function is undefined at a certain point due to both the numerator and the denominator being zero. These are not the same as asymptotes. Asymptotes continue infinitely, while holes are isolated points.

For example, in the function \( h(x)=\frac{x+6}{x^{2}+2x-24} \), at \( x = -6 \), both \( x+6 \) and \( (x+6)(x-4) \) equal zero. Hence, there is a hole at this \( x \) value.

To identify a hole, ensure that the zero cancels out when simplified, indicating a removable discontinuity. Such points do not exist on the actual function graph but should be noted when analyzing the function.

Denominator Factorization

Denominator factorization is a technique used to determine the zeros of the denominator, aiding in finding vertical asymptotes and holes.

For the given function \( h(x)=\frac{x+6}{x^{2}+2x-24} \), we factor the denominator \( x^{2}+2x-24 \). Using the factorization method, we get \( (x-4)(x+6) \).

Solving \( (x-4)(x+6) = 0 \) results in \( x = 4 \) and \( x = -6 \). These values indicate where the denominator becomes zero.

If the numerator is non-zero at these values, then there are vertical asymptotes; if the numerator is zero, it indicates a hole where it cancels out. This highlights the functional points of discontinuity that can reshape and define the rational function graph.