Question

Finding Vertical Asymptotes In Exercises \(13-28\) , find the vertical asymptotes (if any) of the graph of the function. $$ f(x)=\frac{x^{2}-2 x-15}{x^{3}-5 x^{2}+x-5} $$

Short answer

The vertical asymptotes of the function are \(x=1\) and \(x=2\).

Step-by-step solution

Factorize the polynomials

Factorize the polynomials in the numerator and denominator. The numerator \(x^{2}-2 x-15\) becomes \((x-5)(x+3)\), and the denominator \(x^{3}-5 x^{2}+x-5\) becomes \((x-1)(x-2)^{2}\). So, the function can be rewritten as \(f(x)=\frac{(x-5)(x+3)}{(x-1)(x-2)^{2}}\)

Set denominator equal to zero

Now, set the denominator equal to zero. This gives the equation \((x-1)(x-2)^{2}=0\).

Solve for x

Solving this equation for x will involve applying the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. This gives two possible solutions: \(x=1\) and \(x=2\) .

Check the solutions

Now, check these solutions in the original function. The requirement for the vertical asymptotes is that they make the denominator zero, while not making the numerator zero at the same time. Upon checking it can be seen that substituting \(x=1\) or \(x=2\) into the numerator does not result in a zero.

Key concepts

Polynomial Factorization

Polynomial factorization is a crucial first step in finding vertical asymptotes for rational functions. It involves breaking down a polynomial into simpler terms, known as factors, which when multiplied together give the original polynomial. This process simplifies the analysis of the function.
For the given function \(f(x) = \frac{x^{2} - 2x - 15}{x^{3} - 5x^{2} + x - 5}\), the numerator \(x^{2} - 2x - 15\) is factorized into \((x-5)(x+3)\). Similarly, the denominator \(x^{3} - 5x^{2} + x - 5\) is factorized into \((x-1)(x-2)^{2}\).

• Factorization helps to identify the roots of the polynomials, simplifying further calculations.
• It converts a complex polynomial into a product of its factors, making it easier to see where the function might become undefined.

Denominator Zero

A denominator zero indicates a potential vertical asymptote in a rational function. This is because, mathematically, division by zero is undefined, causing the graph to "shoot up" to infinity or "dip down" to negative infinity at certain points.
Once the polynomial is factorized, we set the factors of the denominator equal to zero. This gives us the x-values where the denominator becomes zero, possibly indicating infinite behavior or jumps in the graph.
For the function \(f(x)\), setting \((x-1)(x-2)^{2} = 0\) helps us find where these discontinuities occur.

• This process pinpoints the values that make the function undefined due to division by zero.
• It is essential for understanding where a vertical asymptote might be located.

Zero-Product Property

The zero-product property is pivotal when determining vertical asymptotes since it simplifies determining the roots of polynomial equations. This property asserts that if a product of several factors is zero, at least one of the factors must be zero.
With our factored denominator \((x-1)(x-2)^{2}\), applying the zero-product property allows us to solve \(x-1 = 0\) and \(x-2 = 0\). Consequently, we find the possible x-values for vertical asymptotes: \(x = 1\) and \(x = 2\).

• This property is straightforward and essential for finding the points where the function becomes undefined.
• It is useful for solving any polynomial equation where the objective is to find the values of x that "break" the function's continuity.

Function Graph Analysis

Function graph analysis, specifically related to vertical asymptotes, involves understanding how changes in the function affect its graph, particularly where it might not be defined. When \(x = 1\) or \(x = 2\) results in a zero denominator without a simultaneous zero numerator, vertical asymptotes occur.
In our function \(f(x)\), when substituting \(x = 1\) or \(x = 2\), neither makes the numerator zero, confirming that these values are indeed vertical asymptotes.

• Vertical asymptotes manifest as lines the graph approaches but never crosses.
• Understanding them helps predict and sketch the behavior of the function.

These aspects are crucial for analyzing and interpreting graphs, particularly in rational functions where asymptotic behavior is common.