Problem 49 - 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. \( y = \dfrac{2x^2 + x - 1}{x^2 + x -2} \)

Step-by-Step Solution

Answer

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

Step 1: Determine Vertical Asymptotes

To find vertical asymptotes, set the denominator equal to zero and solve for \( x \). The denominator is \( x^2 + x - 2 \). Set it equal to zero: \[ x^2 + x - 2 = 0 \] This can be factored into: \[ (x-1)(x+2) = 0 \]Setting each factor to zero gives the potential vertical asymptotes solutions: \( x = 1 \) and \( x = -2 \). Verify if they are true vertical asymptotes by ensuring they do not cancel with any factor in the numerator.

Step 2: Define Common Factors

Check if the factors \( (x-1) \) or \( (x+2) \) appear in the numerator \( 2x^2 + x - 1 \). Factor the numerator, if possible, to see if any term cancels: The expression cannot be factored further into linear polynomials, confirming no common factors with the denominator. Thus, vertical asymptotes are \( x = 1 \) and \( x = -2 \).

Step 3: Determine Horizontal Asymptote

For horizontal asymptotes, compare the degrees of the numerator and the denominator. Both the numerator \( 2x^2 + x - 1 \) and the denominator \( x^2 + x - 2 \) are of degree 2. When the degrees are the same, the horizontal asymptote is given by the ratio of the leading coefficients: \( \frac{2}{1} = 2 \).Thus, the horizontal asymptote is \( y = 2 \).

Key Concepts

Horizontal Asymptotes

Horizontal asymptotes give us vital information about the behavior of a graph as it stretches towards infinity. For a rational function, horizontal asymptotes are determined by comparing the degrees of its numerator and its denominator.
Let's break it down:- **Degree Compare**: If the degree of the numerator is smaller than the degree of the denominator, the horizontal asymptote is the x-axis, or \( y = 0 \).
- **Equal Degrees**: If the numerator's degree equals the denominator's degree, the horizontal asymptote equals the ratio of the leading coefficients of the numerator and denominator. Like in our solution, with \( rac{2}{1} = 2 \), resulting in \( y = 2 \).
- **Numerator Greater**: When the numerator's degree is higher than the denominator's, no horizontal asymptote exists.This helps in predicting long-term behavior on a graph and it’s especially important in calculus and analysis.

Vertical Asymptotes

Vertical asymptotes occur at values of \( x \) where the function is undefined, and they often signify a point where the graph shoots up to positive or negative infinity.
Here’s how to find them:- **Denominator Equals Zero**: Start by identifying the points where the denominator of the rational function equals zero. These points might be vertical asymptotes.
- **Factoring**: Factor both numerator and denominator. In our exercise, the factorization shows \((x-1)\) and \((x+2)\) in the denominator, giving potential asymptotes at \( x = 1 \) and \( x = -2 \).
- **Verify**: Ensure that these factors don’t cancel out with any in the numerator.Understanding where vertical asymptotes lie helps to accurately sketch graphs and understand intervals of continuity and discontinuity.

Rational Functions

Rational functions are fractions where both the numerator and the denominator are polynomials. They form the backbone for many concepts in algebra and calculus.
Here's what makes them tick:- **Form and Simplification**: A rational function looks like \( f(x) = \frac{P(x)}{Q(x)} \) where \( P(x) \) and \( Q(x) \) are polynomials. Always check if it can be simplified by canceling common factors between the numerator and the denominator. Beware: canceling wrongly can lead to losing critical asymptote information.
- **Intercepts and Asymptotes**: Determine crucial points like y-intercepts \((f(0))\) and identify all asymptotes, which together give a logical shape to the graph.
- **Graphing**: Armed with horizontal and vertical asymptotes, you can sketch a basic graph. Rational functions often change direction sharply at their asymptotes, indicating infinite limits.Mastering these functions opens doors to understanding much more complex mathematical behaviors!