The image you uploaded includes a problem-solving approach for analyzing rational functions, focusing on aspects like vertical asymptotes (VA), horizontal asymptotes (HA), intercepts, and other properties of the graph such as maxima, minima, and concavity.

Here's a breakdown of the approach presented:

1. Identify Vertical and Horizontal Asymptotes (VA & HA):

   • Vertical asymptotes occur when the denominator of the function equals zero.
   • Horizontal asymptotes describe the behavior of the function as $x \to \infty$.

2. Find Critical Points:

   • These are points where the derivative is zero or undefined and help identify local maxima, minima, and intervals of increase or decrease.
   • Determine where the function changes concavity (inflection points).

3. Sketch the Graph:

   • Based on the information from the previous steps, sketching the graph becomes possible.

Example function in the image:

$f(x) = x + \frac{1}{x}$

Steps provided in the image:

1. Asymptotes and Intercepts:

   • Vertical Asymptote (VA): At $x = 0$.
   • Horizontal Asymptote (HA): None.
   • x-intercept: None.
   • y-intercept: None.

2. Second Function: $f(x) = 1 - \frac{1}{x^2}$

   • The image includes a rough sketch of its behavior, showing that the function has a vertical asymptote at $x = 0$, and positive or negative values on either side, indicating no horizontal asymptote.

If you want to go further into solving these problems or have specific questions on one of these steps, feel free to ask! Would you like a detailed breakdown of a particular step or more examples?

Here are five related questions that might expand the topic:

1. How do you find the vertical asymptotes of rational functions in general?
2. What is the significance of horizontal asymptotes in graphing rational functions?
3. How do you determine if a function is increasing or decreasing on an interval?
4. What is the process for finding inflection points in a rational function?
5. How can we analyze a function's concavity using its second derivative?

Tip: Always factor the denominator first to check for vertical asymptotes when analyzing rational functions!