Problem 46 - 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. \(F(x)=\frac{3}{9-x^{2}}\)

Step-by-Step Solution

Answer

Vertical asymptotes: \(x = 3\) and \(x = -3\). Horizontal asymptote: \(y = 0\).

Step 1: Identify Vertical Asymptotes

Vertical asymptotes occur where the denominator of the function is equal to zero, and cannot be canceled by the numerator. For the function \(F(x)=\frac{3}{9-x^{2}}\), we need to solve for \(x\) where \(9-x^{2}=0\). Solving this gives: \(-x^2 = -9\) or \(x^2 = 9\). Hence, the solutions are \(x = 3\) and \(x = -3\). Therefore, the vertical asymptotes are \(x = 3\) and \(x = -3\).

Step 2: Identify Horizontal Asymptotes

Horizontal asymptotes are found by looking at the degrees of the numerator and the denominator. The function \(F(x)=\frac{3}{9-x^{2}}\) has a numerator with degree 0 and a denominator with degree 2. Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \(y = 0\).

Step 3: Sketch the Graph

Based on the asymptotes found, sketch the graph. The graph approaches the vertical asymptotes at \(x=3\) and \(x=-3\) vertically. As \(x\) tends towards infinity or negative infinity, the graph approaches the horizontal asymptote at \(y=0\). The function is positive over its domain except between \(x=-3\) and \(x=3\), where the denominator would be negative.

Key Concepts

Vertical Asymptotes

Vertical asymptotes are lines that the graph of a function approaches but never touches or crosses. They occur when the denominator of a rational function equals zero, causing the function to be undefined at that point.
To find them for a function like \( F(x) = \frac{3}{9-x^2} \), you need to set the denominator equal to zero and solve for \( x \). In this case, we solve \( 9 - x^2 = 0 \), which simplifies to \( x^2 = 9 \). Taking the square roots, we find \( x = 3 \) and \( x = -3 \). These are locations where the function heads towards positive or negative infinity as it approaches these points from either side.
Keep in mind that vertical asymptotes do not exist where factors in the numerator also cancel out zero-value points in the denominator.

Horizontal Asymptotes

Horizontal asymptotes show the behavior of a function as \( x \) moves towards positive or negative infinity. They don't indicate values that the function can never reach, but rather the general direction it levels off.
For the function \( F(x) = \frac{3}{9-x^2} \), to determine the horizontal asymptote, we look at the degrees of the numerator and denominator. The numerator has a lower degree (0) than the denominator (2). This means, generally, as \( x \to \infty \) or \( x \to -\infty \), the function approaches \( y = 0 \).
It's crucial to understand that horizontal asymptotes describe end-behavior, so while the graph may cross this horizontal line, it will always "flatten out" close to it as \( x \) grows large in magnitude.

Graph Sketching

Sketching the graph of a function involves combining information about asymptotes and other function behaviors. The visual representation helps clarify how the function behaves around different \( x \) values.
Start by plotting the vertical asymptotes you previously identified. For \( F(x) = \frac{3}{9-x^2} \), these should be at \( x = 3 \) and \( x = -3 \). Draw dashed lines here to represent points of non-existence.
Then, hint at the horizontal asymptote at \( y = 0 \). As \( x \) becomes very large or very small, the graph will come close to this line. Note the sign of the function in various intervals. Since the function is positive except between \( x = -3 \) and \( x = 3 \), expect it to be above the \( x \)-axis outside these bounds, and spread below when between the asymptotes.

• Check for any intercepts to give more structure to your sketch.
• Consider testing key points to understand further how the function behaves around the asymptotes and intercepts.

Graph sketches are extremely useful for visualizing the complex interplay between these mathematical concepts.