Problem 47 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the horizontal and vertical asymptotes for the graphs of the indicated functions. Then sketch their graphs. \(g(x)=\frac{14}{2 x^{2}+7}\)

Step-by-Step Solution

Answer

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

Step 1: Identify the horizontal asymptote

Horizontal asymptotes are considered by observing the degrees of the polynomial in the numerator and the polynomial in the denominator. Here, the degree of the numerator is 0 (since it is a constant) and the degree of the denominator is 2. Since the degree of the denominator is greater, the horizontal asymptote is at \(y = 0\).

Step 2: Identify the vertical asymptotes

Vertical asymptotes occur where the function is undefined, which happens where the denominator is zero. So, set the denominator equal to zero and solve for \(x\): \[2x^2 + 7 = 0\] Since \(2x^2 + 7\) cannot be zero (as it can't equal a negative number or zero when squared and multiplied by 2), there are no real solutions. Thus, there are no vertical asymptotes.

Step 3: Sketch the graph

Start by plotting the horizontal asymptote at \(y = 0\). Since there are no vertical asymptotes, plot points to ensure the graph correctly approaches the horizontal asymptote. Notice that as \(x\) approaches positive or negative infinity, the function \(g(x)\) approaches zero. The function remains positive for all real \(x\), resulting in a curve above the horizontal asymptote that decreases towards zero as \(x\) increases or decreases.

Key Concepts

Horizontal Asymptotes

Horizontal asymptotes help us understand the behavior of a function as it moves towards the far ends of the x-axis, either toward positive or negative infinity. For the function given, \[g(x) = \frac{14}{2x^2 + 7},\]the degree of the numerator is zero because the numerator is a constant, while the degree of the polynomial in the denominator is two. When the degree of the denominator is greater than the degree of the numerator, as it is here, the horizontal asymptote is given by \(y = 0\). This is similar to saying that as \(x\) becomes extremely large or extremely small, the value of \(g(x)\) gets closer and closer to zero.

Key points to remember about horizontal asymptotes:

• If the numerator's degree is less than the denominator's, \(y = 0\) is the asymptote.
• If the numerator's degree equals the denominator's, the asymptote is the ratio of the leading coefficients.
• If the numerator's degree is greater than the denominator's, there is no horizontal asymptote.

Vertical Asymptotes

Vertical asymptotes represent points where a function becomes undefined and the graph heads towards infinity in either direction. This occurs when the denominator of a rational function is zero. However, for our function, \[g(x) = \frac{14}{2x^2 + 7},\]no real \(x\) value will make the denominator zero. To find vertical asymptotes, you normally solve the equation: \[2x^2 + 7 = 0.\]Unfortunately, this equation has no real solutions because a squared term is always greater than zero, and adding a positive number (7) only increases it.

Therefore, the denominator does not equal zero for any real \(x\). Consequently, there are no vertical asymptotes for this function.

• Vertical asymptotes are located where the denominator equals zero and the numerator is non-zero.
• A vertical asymptote signifies infinite growth or decay of the graph at those points.

Polynomial Functions

In the realm of calculus and algebra, polynomial functions often play a central role. These are mathematical expressions involving variables and coefficients, structured through the operations of addition, multiplication, and non-negative integer exponents. For our function, the denominator is a polynomial: \[2x^2 + 7.\] Here, the highest exponent is 2, making it a quadratic polynomial.

Polynomials are generally categorized by their degree:

• Degree 0: Constant functions, like \(14\), which doesn't change.
• Degree 1: Linear functions, such as \(2x + 3\), forming lines.
• Degree 2: Quadratic functions, like \(2x^2 + 7\), creating parabolic curves.

For the analysis of rational functions like \(g(x)\), understanding the polynomial degree in both, the numerator and the denominator, is essential for determining asymptotic behaviors. Polynomial functions, by their nature, provide a predictable path in terms of their growth, symmetry, and zeros, making them reliable and essential tools in calculating limits and asymptotes.