Problem 44 - College Algebra and Calculus: An Applied Approach

Question

Sketch the graph of the rational function. To aid in sketching the graphs, check for intercepts, symmetry, vertical asymptotes, and horizontal asymptotes. $$P(x)=\frac{1-3 x}{1-x}$$

Step-by-Step Solution

Answer

The y-intercept is at (0, 1). There are no x-intercepts. The function is neither even nor odd. The vertical asymptote is at x = 1, and the horizontal asymptote is y = 3. The graph approaches these asymptotes but never crosses them.

Step 1: Find the Intercepts

To find the x-intercept, set \(P(x) = 0\) and solve for x. For the y-intercept, set x = 0 and solve for \(P(x)\)

Step 2: Check for Symmetry

Check if the function is even, odd, or neither. A function is even if \(P(-x) = P(x)\), odd if \(P(-x) = -P(x)\), and neither if it doesn't meet either condition.

Step 3: Identify Vertical Asymptotes

Vertical asymptotes occur when the denominator of a rational function is 0. Set the denominator \(1 - x = 0\), and solve for x to find the vertical asymptote.

Step 4: Identify Horizontal Asymptotes

The horizontal asymptote is found by examining the degrees of the polynomial in the numerator and the denominator. If the degrees are the same, the horizontal asymptote is the ratio of the coefficients of the highest degree terms. If the degree of the numerator is less than the denominator, the asymptote is y = 0. If the degree of the numerator is greater than the denominator, there is no horizontal asymptote.

Step 5: Sketch the Graph

Using the intercepts, symmetry, and asymptotes identified in the previous steps, sketch the graph of the function.

Key Concepts

Vertical Asymptotes

Vertical asymptotes are like invisible barriers that the graph of a function approaches but never touches or crosses. They represent values that a function cannot equal and typically occur where the denominator of a rational function is zero, leading to undefined points. For example, let's examine the rational function \(P(x) = \frac{1-3x}{1-x}\). To find the vertical asymptote, we set the denominator equal to zero: \(1 - x = 0\). Solving for x gives us \(x = 1\). Thus, \(x = 1\) is our vertical asymptote for this function. When graphing, we draw a dashed vertical line at \(x = 1\), and this serves as a guide to indicate that the function will approach this line infinitely as x approaches 1 from either side.

When preparing to graph a rational function, it is crucial to identify vertical asymptotes early because they significantly influence the shape and direction of the graph.

Horizontal Asymptotes

On the other hand, horizontal asymptotes indicate the behavior of a function as the x-values head towards positive or negative infinity. These are the constant values that the function approaches at the extreme ends of the x-axis but does not necessarily reach. To find the horizontal asymptote for the given function \(P(x) = \frac{1-3x}{1-x}\), we observe the degrees of the numerator and the denominator. Both are first-degree polynomials. Since the degrees are the same, the horizontal asymptote is determined by the ratio of the leading coefficients. Here, we have \(-3\) and \(-1\), respectively, giving us a horizontal asymptote of \(y = \frac{-3}{-1} = 3\). We draw a dashed horizontal line at \(y = 3\) on the graph, which represents the value the function approaches as x tends toward infinity.

Knowing the horizontal asymptote is essential for understanding the long-term behavior of a function and is a significant aspect of its graphical representation.

X-Intercepts and Y-Intercepts

X-intercepts and y-intercepts are points where the graph of the function crosses the x-axis and y-axis, respectively. They provide crucial information about the function's behavior and help create a rough sketch of the graph. To find the x-intercepts for the function \(P(x) = \frac{1-3x}{1-x}\), we set the function equal to zero: \(0 = \frac{1-3x}{1-x}\) and solve for x. Upon performing the calculation, we find that the x-intercept occurs at \(x = \frac{1}{3}\). For the y-intercept, we set x equal to zero within the function and solve for \(P(x)\), which yields \(y = 1\). These intercepts are plotted on the graph and give a basic framework for the function's curve.

Understanding where and how to calculate these intercepts is vital for graphing rational functions accurately and provides a starting point for the entire graphing process.

Function Symmetry

Function symmetry can greatly simplify the process of graphing. A function is symmetric if it shows reflective symmetry across the y-axis (even symmetry), symmetry about the origin (odd symmetry), or has no symmetry. To determine the symmetry of the function \(P(x) = \frac{1-3x}{1-x}\), we replace \(x\) with \(-x\) to get \(P(-x) = \frac{1-3(-x)}{1-(-x)}\). Simplifying gives us \(P(-x) = \frac{1+3x}{1+x}\), which is not equal to \(P(x)\) nor \(-P(x)\). Therefore, the function has no symmetry. Symmetry informs us about the function's behavior and aids in predicting its graph. When a function has symmetry, less work is required as you can reflect the points and shape across the axis of symmetry to complete the graph.

Understanding symmetry not only saves time but also prevents potential mistakes during the graphing process, making it a powerful tool in the study of rational functions.