Problem 18 - A Graphical Approach to Precalculus with Limits

Question

For function find all asymptotes and the coordinates of any holes in its graph. $$f(x)=\frac{3 x^{2}+x-1}{2 x^{2}+3 x-2}$$

Step-by-Step Solution

Answer

Vertical asymptotes at \(x = \frac{1}{2}\) and \(x = -2\); horizontal asymptote at \(y = \frac{3}{2}\); no holes.

Step 1: Factor the Numerator and Denominator

The first step in finding asymptotes and holes is to factor both the numerator and the denominator of the function. We have:\[f(x) = \frac{3x^2 + x - 1}{2x^2 + 3x - 2}\]For the numerator, factor as follows:\[3x^2 + x - 1 = (3x - 1)(x + 1)\]For the denominator, factor as follows:\[2x^2 + 3x - 2 = (2x - 1)(x + 2)\]

Step 2: Identify Holes

Holes occur where a factor is common in both the numerator and denominator and cancels out. In this case, the factors do not simplify down further, as there are no common factors between the numerator and denominator. Thus, there are no holes in this function.

Step 3: Find Vertical Asymptotes

Vertical asymptotes occur when the denominator is zero and the numerator is not zero at the same points. Set the factors of the denominator equal to zero:\[2x - 1 = 0 \quad \Rightarrow \quad x = \frac{1}{2}\]\[x + 2 = 0 \quad \Rightarrow \quad x = -2\]Thus, the vertical asymptotes are at \(x = \frac{1}{2}\) and \(x = -2\).

Step 4: Find Horizontal Asymptotes

Horizontal asymptotes are determined by the degrees of the polynomials. Here, both the numerator and denominator are quadratic (degree 2). The horizontal asymptote can be found using the ratio of the leading coefficients. Thus, the horizontal asymptote is:\[y = \frac{3}{2}\]

Key Concepts

Holes in Rational Functions

Rational functions sometimes have holes, which are points on the graph where the function isn't defined. These occur when a factor is common to both the numerator and denominator. When these common factors are canceled out, they leave a "hole". To determine if a function has a hole, first factor both the numerator and the denominator. After factoring, check for any common factors. If they exist, set the common factor equal to zero to find the x-coordinate of the hole. Substitute this x-coordinate back into the reduced form of the function to find the corresponding y-coordinate. For the function in this problem, given by \( f(x) = \frac{3x^2 + x - 1}{2x^2 + 3x - 2} \), we've factored both the numerator and the denominator, but found no common factors. This means there are no holes in the graph of this function.

Vertical Asymptotes

Vertical asymptotes occur in a rational function when the value of x makes the denominator zero, while the numerator is not zero at the same x-value. To find vertical asymptotes, solve for the values of x that make the denominator equal to zero and ensure these do not also make the numerator zero.
In this problem, the factored form of the denominator is \((2x - 1)(x + 2)\). Set each factor to zero to find the vertical asymptotes:

• For \(2x - 1 = 0\), solving gives \(x = \frac{1}{2}\).
• For \(x + 2 = 0\), solving gives \(x = -2\).

Neither x-value makes the numerator zero, so these are indeed vertical asymptotes. Thus, the function has vertical asymptotes at \(x = \frac{1}{2}\) and \(x = -2\). These asymptotes indicate where the graph will shoot up to infinity or down to negative infinity.

Horizontal Asymptotes

Horizontal asymptotes are identified by comparing the degrees of the numerator and the denominator in a rational function.

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \(y = 0\).
• If they are equal, the horizontal asymptote is \( y = \frac{a}{b} \) where \(a\) and \(b\) are the leading coefficients of the numerator and the denominator, respectively.
• If the numerator's degree is greater, there is no horizontal asymptote, but possibly an oblique one.

For the function \(f(x) = \frac{3x^2 + x - 1}{2x^2 + 3x - 2}\), both the numerator and denominator have degree 2. Therefore, the horizontal asymptote is determined by the leading coefficients: \(\frac{3}{2}\). Hence, the horizontal asymptote is \(y = \frac{3}{2}\). Horizontal asymptotes give us valuable information about the behavior of the graph as \(x\) approaches infinity or negative infinity.