Problem 35 - College Algebra with Modeling and Visualization

Question

Let a be a positive constant. Match \(f(x)\) with its graph \((a-d)\) without using a calculator. $$ f(x)=\frac{x-a}{x+2} $$

Step-by-Step Solution

Answer

The graph has a vertical asymptote at \( x = -2 \), a horizontal asymptote at \( y = 1 \), and an x-intercept at \( x = a \).

Step 1: Analyze the Structure of the Function

The function given is \( f(x) = \frac{x-a}{x+2} \). This is a rational function, meaning it is the quotient of two polynomials. The numerator is \( x-a \) and the denominator is \( x+2 \). Since both are linear functions, the function represents a rational graph with potential asymptotes and intercepts.

Step 2: Identify the Vertical Asymptote

The vertical asymptote occurs where the denominator is zero. So, set the denominator equal to zero: \( x + 2 = 0 \). Solve for \( x \): \( x = -2 \). Thus, there is a vertical asymptote at \( x = -2 \).

Step 3: Determine the Horizontal Asymptote

For rational functions where the degrees of the numerator and denominator are equal, the horizontal asymptote is found by dividing the leading coefficients. Here, both the numerator and denominator have the same degree 1, hence the horizontal asymptote is \( y = 1 \).

Step 4: Find the x-intercept

The x-intercept occurs where \( f(x) = 0 \). Set the numerator equal to zero: \( x-a = 0 \). Solve for \( x \): \( x = a \). Therefore, the x-intercept is at \( (a, 0) \).

Step 5: Evaluate the Graph's Behavior Near Asymptotes

As \( x \) approaches \(-2\), the function approaches infinity or negative infinity - behavior changes according to signs of factors. When \( x \) is large (positively or negatively), the function approaches the horizontal asymptote \( y = 1 \), indicating the ends of the graph flatten out toward \( y = 1 \).

Key Concepts

Vertical Asymptotes

Vertical asymptotes occur in rational functions where the denominator equals zero, causing the function value to approach infinity and create a division by zero scenario. In our function, \( f(x) = \frac{x-a}{x+2} \), the denominator is \( x + 2 \).

To find the location of a vertical asymptote, set the denominator equal to zero and solve for \( x \):

• \( x + 2 = 0 \)
• \( x = -2 \)

There is a vertical asymptote at \( x = -2 \). As \( x \) gets closer to \(-2\), the function values swing towards positive or negative infinity. Hence, the graph will have a vertical line at \( x = -2 \), representing this boundary.

Understanding where vertical asymptotes are can help you visualize the function's behavior and understand the restrictions on \( x \). They are important in analyzing how the function behaves as it nears the undefined points.

Horizontal Asymptotes

Horizontal asymptotes describe how the function behaves as \( x \) approaches positive or negative infinity. For rational functions where the numerator and denominator have the same degree, the horizontal asymptote can be determined by their leading coefficients.

In the function \( f(x) = \frac{x-a}{x+2} \), both the numerator \( (x-a) \) and the denominator \( (x+2) \) are of the same degree, which is 1. Therefore, to find the horizontal asymptote, divide the leading coefficients:

• Numerator's leading coefficient: 1
• Denominator's leading coefficient: 1

Resulting in a horizontal asymptote at \( y = 1 \). This asymptote suggests that, as \( x \) becomes very large (either positively or negatively), the value of \( f(x) \) approaches 1.

Horizontal asymptotes give a clearer picture of the function's behavior at infinity and tell you where the graph will level off. This insight is crucial in graph interpretation.

X-Intercepts

X-intercepts are the points where the graph of a function crosses the x-axis. For a rational function, this occurs when the numerator equals zero, because the entire function evaluates to zero.

In \( f(x) = \frac{x-a}{x+2} \), find the x-intercept by setting the numerator \( x-a \) to zero:

• \( x-a = 0 \)
• \( x = a \)

Thus, the x-intercept of the function is at the point \( (a, 0) \). This means when \( x = a \), the function value is 0, and the graph crosses the x-axis.

Recognizing x-intercepts helps in quickly sketching the graph of a function. Intercepts serve as checkpoints for ensuring the plot's accuracy and understanding the function's roots.