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.
y = (x² - 49) / (x² + 5x - 14)

Verified step by step guidance

1. Step 1: Analyze the function's domain. The denominator of the function is \(x^2 + 5x - 14\). Set the denominator equal to zero to find the values of \(x\) that make the function undefined. Solve \(x^2 + 5x - 14 = 0\) using factoring or the quadratic formula to determine the vertical asymptotes.
2. Step 2: Determine the horizontal asymptote by analyzing the degrees of the numerator \(x^2 - 49\) and the denominator \(x^2 + 5x - 14\). Since the degrees of the numerator and denominator are the same, the horizontal asymptote is given by the ratio of the leading coefficients.
3. Step 3: Find the critical points by taking the derivative of \(y = \frac{x^2 - 49}{x^2 + 5x - 14}\). Use the quotient rule: \(\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2}\), where \(u = x^2 - 49\) and \(v = x^2 + 5x - 14\). Set the derivative equal to zero to find the values of \(x\) where the slope of the tangent is zero.
4. Step 4: Identify inflection points by finding the second derivative of the function. Use the quotient rule again to compute the second derivative, and set it equal to zero to find the values of \(x\) where the concavity changes.
5. Step 5: Evaluate the function at the critical points, inflection points, and endpoints (if applicable) to determine the coordinates of local extreme points, inflection points, and absolute extreme points. Use the information gathered to graph the function, marking all key features such as asymptotes, intercepts, and extrema.

Key Concepts

Graphing Rational Functions

Graphing rational functions involves analyzing the function's behavior by identifying key features such as intercepts, asymptotes, and critical points. The function given, y = (x² - 49) / (x² + 5x - 14), can be graphed by determining where the numerator and denominator equal zero, which helps locate x-intercepts and vertical asymptotes.

Local and Absolute Extrema

Local extrema refer to points where a function reaches a maximum or minimum value within a specific interval, while absolute extrema are the highest or lowest points over the entire domain. To find these points, one typically uses the first derivative test to identify critical points and the second derivative test to determine their nature.

Inflection Points

Inflection points occur where the concavity of a function changes, which can be identified by analyzing the second derivative. For the function y = (x² - 49) / (x² + 5x - 14), finding inflection points involves setting the second derivative equal to zero and solving for x, indicating where the graph changes from concave up to concave down or vice versa.