Problem 64 - College Algebra

Question

a. Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. $$f(x)=\frac{x^{2}-x+1}{x-1}$$

Step-by-Step Solution

Answer

The equation of the slant asymptote is \(y = x\). The graph of the function approaches this line as \(x\) tends to positive or negative infinity. The function also has an x-intercept at \(x=1\), a y-intercept at \(y=0\) and a vertical asymptote at \(x=1\).

Step 1: Polynomial Division

Apply polynomial division on the given function to get the expression that will represent the slant asymptote. Divide \(x^{2}-x+1\) by \(x-1\).

Step 2: Find the Slant Asymptote

The expression you obtained in Step 1 is the slant asymptote. This equation represents a line that the curve approaches as \(x\) approaches to positive or negative infinity.

Step 3: Strategy for Graphing

Now to graph the function, start by sketching the slant asymptote from Step 2. Then, identify the x-intercepts by setting the numerator equal to zero and solving for \(x\). Similarly, identify the y-intercept by setting \(x = 0\) in the original function. Lastly, note down any vertical asymptotes by setting the denominator equal to zero and solving for \(x\).

Step 4: Finalizing the Graph

Using all this information from Step 3, plot the asymptotes and x and y intercepts. As \(x\) tends to infinity in the positive and negative direction, the graph of the function will approach the slant asymptote. Then use those plotted points and tendencies to sketch the graph of the full function.

Key Concepts

Slant Asymptote

In graphing rational functions, a slant asymptote, also known as an oblique asymptote, is a line the graph approaches as the independent variable, usually noted as x, heads towards infinity or negative infinity. Unlike horizontal asymptotes, slant asymptotes occur when the degree (the highest exponent) of the polynomial in the numerator is exactly one more than the degree of the polynomial in the denominator.

To find the slant asymptote of a function, we use polynomial division to divide the numerator by the denominator. The quotient (without the remainder) gives the equation of the slant asymptote. For the given function f(x) = (x^2 - x + 1) / (x - 1), the division yields a linear function which is our slant asymptote. As x approaches large positive or negative values, the graph of the function will get ever closer to this line, though it will never actually touch it.

Polynomial Division

The concept of polynomial division is analogous to long division that is performed with numbers. It is used to simplify expressions and find slant asymptotes in rational functions. Polynomial division helps to break down a complex rational function into a more simple form that is easier to understand and graph.

When performing polynomial division, we arrange the terms of the polynomials in decreasing order of their degrees and divide the leading term of the numerator by the leading term of the denominator to find the first term of the quotient. This process continues until all terms of the numerator are divided, resulting in a quotient representing the slant asymptote and a remainder that becomes less significant as x grows.

X-Intercepts

The x-intercepts of a graph represent the points where the curve crosses the x-axis. In other words, they are the values of x for which the function f(x) equals zero. To find the x-intercepts algebraically, we set the numerator of the rational function to zero and solve for x. If the numerator can be factored, this may involve finding the roots of the factored expression.

For instance, with our function f(x), setting the numerator x^2 - x + 1 to zero would typically help us find the x-intercepts. However, in some cases, as in this function, the equation might not have real solutions, indicating that the graph doesn't cross the x-axis at any point.

Y-Intercepts

A graph's y-intercept is the point where it intersects the y-axis, which corresponds to the value of the function when x is zero. To find the y-intercept, we substitute x with zero in the original function and solve for f(0).

In the context of the given function f(x) = (x^2 - x + 1) / (x - 1), setting x to zero produces the y-intercept at the point (0, f(0)). It's essential for graphing, as it provides a specific point through which the curve will pass, serving as an anchor point for constructing the rest of the graph.