In Exercises 9–66, graph the function using appropriate methods from the graphing procedures presented just before Example 9, identifying the coordinates of any local extreme points and inflection points. Then find coordinates of absolute extreme points, if any.
33. y = (x² - x + 1) / (x - 1)

Verified step by step guidance

1. Step 1: Begin by analyzing the function y = (x² - x + 1) / (x - 1). Identify any points where the function is undefined. The denominator x - 1 is zero when x = 1, indicating a potential vertical asymptote at x = 1.
2. Step 2: Determine the behavior of the function as x approaches the vertical asymptote from both sides. Evaluate the limits as x approaches 1 from the left and right to understand the behavior near the asymptote.
3. Step 3: Find the first derivative of the function to identify local extreme points. Use the quotient rule: if y = u/v, then y' = (u'v - uv')/v². Apply this to y = (x² - x + 1)/(x - 1) to find y'. Set y' = 0 to find critical points.
4. Step 4: Calculate the second derivative to find inflection points. Use the derivative obtained in Step 3 and differentiate again. Set the second derivative equal to zero to find potential inflection points.
5. Step 5: Evaluate the function at critical points and endpoints (if any) to find absolute extreme points. Compare the values of the function at these points to determine the absolute maximum and minimum values.

Key Concepts

Graphing Rational Functions

Graphing rational functions involves identifying key features such as asymptotes, intercepts, and behavior at infinity. For the function y = (x² - x + 1) / (x - 1), vertical asymptotes occur where the denominator is zero, and horizontal asymptotes are determined by the degrees of the numerator and denominator. Understanding these features helps in sketching the graph accurately.

Local Extreme Points

Local extreme points are points where the function reaches a local maximum or minimum. These can be found by taking the derivative of the function and setting it to zero to find critical points. Analyzing the second derivative or using the first derivative test helps determine the nature of these points, whether they are maxima, minima, or neither.

Inflection Points

Inflection points occur where the function changes concavity, which can be identified by analyzing the second derivative. For y = (x² - x + 1) / (x - 1), finding where the second derivative equals zero or is undefined helps locate these points. Inflection points are crucial for understanding the overall shape and behavior of the graph.