Problem 48 - Calculus: Early Transcendentals

Question

Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes. \( y = \dfrac{2x^2 + 1}{3x^2 + 2x -1} \)

Step-by-Step Solution

Answer

Horizontal asymptote: \( y = \frac{2}{3} \); Vertical asymptotes: \( x = \frac{2}{3} \) and \( x = -1 \).

Step 1: Identify the horizontal asymptote

To find the horizontal asymptote of the function, compare the degrees of the numerator and denominator. Both the numerator and the denominator have the highest degree of 2.

Step 2: Calculate horizontal asymptote equation

Since the degrees of the numerator and denominator are equal, divide the leading coefficients: leading coefficient of the numerator is 2, and the leading coefficient of the denominator is 3. Hence, the horizontal asymptote is \( y = \frac{2}{3} \).

Step 3: Find vertical asymptotes

To find vertical asymptotes, set the denominator equal to zero and solve for \( x \). Thus, solve \( 3x^2 + 2x - 1 = 0 \).

Step 4: Solve the quadratic equation

Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) with \( a = 3, b = 2, c = -1 \). Calculate the discriminant: \( b^2 - 4ac = 4 + 12 = 16 \); solve for \( x \): \( x = \frac{-2 \pm 4}{6} \). This results in \( x = \frac{2}{3} \) and \( x = -1 \). These are the vertical asymptotes.

Key Concepts

Horizontal Asymptotes

Horizontal asymptotes in rational functions provide an idea of how the function behaves as the variable approaches infinity or negative infinity. They are a type of line that the graph of the function gets closer to as you observe larger values of the independent variable, typically denoted as \( x \).

To identify a horizontal asymptote, you should compare the degrees of the polynomial in the numerator and the denominator of the rational function.

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \( y = 0 \).
• If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients.
• If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote (though there might be an oblique asymptote).

In the example \( y = \frac{2x^2 + 1}{3x^2 + 2x - 1} \), both polynomials in the numerator and denominator have a degree of 2. Thus, you find the horizontal asymptote by dividing the leading coefficients: 2 in the numerator and 3 in the denominator, which gives \( y = \frac{2}{3} \).

Vertical Asymptotes

Vertical asymptotes occur when a function approaches an undefined value as \( x \) approaches some constant value. These asymptotes typically represent a value for \( x \) where the rational function approaches infinity or negative infinity.

To find vertical asymptotes, look at the denominator of the function.

• Set the denominator equal to zero.
• Solve the resulting equation for \( x \).

Consider the given example \( y = \frac{2x^2 + 1}{3x^2 + 2x - 1} \). Here, the denominator is \( 3x^2 + 2x - 1 \). Solving \( 3x^2 + 2x - 1 = 0 \) gives us the x-values where vertical asymptotes occur. The quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) with \( a = 3, b = 2, \) and \( c = -1 \) leads to \( x = \frac{-2 \pm 4}{6} \), which simplifies to \( x = \frac{2}{3} \) and \( x = -1 \). These are the vertical asymptotes.

Rational Functions

Rational functions are quotients of two polynomials. They take the form \( \frac{p(x)}{q(x)} \) where both \( p(x) \) and \( q(x) \) are polynomial functions.

The character of these functions largely depends on the degrees of the numerator and the denominator:

• They might not be defined at certain points, those points are typically where the denominator is zero, leading to vertical asymptotes.
• The end behavior of the rational function, or how it acts as \( x \) grows large or small, will often be described by its horizontal asymptotes.
• Sometimes, intercepts or oblique asymptotes could occur if the degree of the numerator exceeds that of the denominator.

Understanding these elements—degrees and coefficients—gives insight into sketching and analyzing how the graph of the rational function behaves.

Quadratic Equations

Quadratic equations are polynomial equations of degree two, typically taking the form \( ax^2 + bx + c = 0 \). The solution to these equations can be found using different methods such as factoring, completing the square, or using the quadratic formula.

In our asymptote exercise, solving the quadratic \( 3x^2 + 2x - 1 = 0 \) helps to find where vertical asymptotes occur. The widely used quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) is a reliable method when other simpler methods aren't feasible.

Key points when using the quadratic formula:

• The expression under the square root, \( b^2 - 4ac \), is called the discriminant and determines the nature of the roots:
• If \( b^2 - 4ac > 0 \), there are two distinct real roots.
• If \( b^2 - 4ac = 0 \), there is exactly one real root.
• If \( b^2 - 4ac < 0 \), there are no real roots (but two complex roots).

For the example of vertical asymptotes, the discriminant \( 16 \) indicates two real roots: \( x = \frac{2}{3} \) and \( x = -1 \).