Problem 24 - Precalculus Mathematics for Calculus

Question

Find all horizontal and vertical asymptotes (if any). $$r(x)=\frac{2 x-4}{x^{2}+x+1}$$

Step-by-Step Solution

Answer

Horizontal asymptote: \( y = 0 \); No vertical asymptotes.

Step 1: Identify Horizontal Asymptotes

For horizontal asymptotes, compare the degrees of the polynomial in the numerator and the denominator. In the function \( r(x) = \frac{2x-4}{x^2+x+1} \), the highest power in the numerator is 1 and in the denominator is 2. Since the degree of the denominator is greater, the horizontal asymptote is \( y = 0 \).

Step 2: Identify Vertical Asymptotes

To find vertical asymptotes, set the denominator equal to zero and solve for \( x \). For \( r(x) = \frac{2x-4}{x^2+x+1} \), solve the equation \( x^2 + x + 1 = 0 \). The discriminant \( b^2 - 4ac = 1^2 - 4 \times 1 \times 1 = -3 \) is negative, indicating no real roots. Therefore, there are no vertical asymptotes.

Key Concepts

Horizontal Asymptotes

Horizontal asymptotes are lines that a graph approaches as the input value, or "x," heads towards positive or negative infinity. In simple terms, they tell us the behavior of a function as we move far away in either direction along the x-axis. For rational functions, such as the one given \[ r(x) = \frac{2x - 4}{x^2 + x + 1},\]horizontal asymptotes are identified by comparing the degrees of the polynomials in the numerator and denominator.

• If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.
• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at \( y = 0 \).
• If the degrees are the same, the horizontal asymptote is found by dividing the leading coefficients.

For our function, the numerator is of degree 1, and the denominator is of degree 2. As the degree of the numerator is less, it confirms that the horizontal asymptote is \( y = 0 \). The graph flattens out to this line as \( x \to \pm\infty \).

Vertical Asymptotes

Vertical asymptotes occur where a function skyrockets to infinity or plummets to negative infinity. They look like vertical lines on a graph and show where the function is undefined because of division by zero. To locate vertical asymptotes in a rational function like \[ r(x) = \frac{2x - 4}{x^2 + x + 1},\]we need to determine where the denominator equals zero, as these are the potential spots for infinite values. Set the denominator equal to zero:\[ x^2 + x + 1 = 0.\]However, when you attempt to solve this using the quadratic formula, you find that the discriminant, which is \( b^2 - 4ac \), equals -3—indicating no real solutions. This means there are no values for \( x \) that cause the denominator to be zero. Thus, no vertical asymptotes exist for this function, and the graph doesn't have vertical boundaries where it "explodes".

Rational Functions

Rational functions are fractions made up of polynomials. For instance, the function \[ r(x) = \frac{2x - 4}{x^2 + x + 1}\]is a rational function because both its numerator and denominator are polynomials. These functions are interesting because they can behave differently in terms of their shape and symmetry, often having characteristic features like asymptotes and intercepts. Understanding their structure helps in predicting how they will look when graphed. Evaluating polynomial degrees in both numerator and denominator gives insights into long-term behavior (horizontal asymptotes) and potential undefined regions (vertical asymptotes). These concepts together help to sketch a rational function efficiently without relying entirely on plotting point by point.

Polynomial Degrees

The degree of a polynomial is the highest power of the variable present in the polynomial. This concept applies directly to understanding asymptotes in rational functions, such as the one shown: \[ r(x) = \frac{2x - 4}{x^2 + x + 1}.\]Understanding Polynomial Degrees:

• In the numerator, \(2x - 4\), the degree is 1 because the highest power is \(x^1\).
• For the denominator, \(x^2 + x + 1\), the degree is 2 as the highest power is \(x^2\).

The polynomial degree plays a crucial role in determining the function's asymptotic behavior. As seen, when the degree of the denominator is higher than the numerator, the horizontal asymptote becomes \( y = 0 \). Meanwhile, exploring the polynomial within the denominator can help indicate potential vertical asymptotes, though a non-real result like what we found means none exist in this case.