use the asymptotes and intercepts to make a quick sketch of the function f(x)= x-2 (numerator) x+4(denominator) and its reciprocal f^r = x+4 (numerator) x-2 (denominator) on the same set of axes.

b) describe the symmetry in the graphs in part a)
c) determine the intervals of increase and decrease for both f and f^r, how do the sets of intervals compare?
d) would the pattern occur for all pairs of functions f(x)= x+b (numerator) x+d and f^r(x) = x+d (numerator)and x+b (denominator)? explain why or why not?

so we have

f(x) = (x-2)/(x+4)
and
g(x) = (x+4)/(x-2)

Using one of my favourite websites:
Wolfram
http://www.wolframalpha.com/input/?i=plot+%28x-2%29%2F%28x%2B4%29%3B%28x%2B4%29%2F%28x-2%29

You should be able to answer your questions from the graph of the two functions

To make a quick sketch of the functions f(x) and f^r (reciprocal), we can analyze the asymptotes and intercepts of both functions.

a) Asymptotes and Intercepts:
1. For f(x) = (x - 2) / (x + 4):
- Vertical Asymptote: Set the denominator equal to zero and solve for x.
x + 4 = 0
x = -4
So, there is a vertical asymptote at x = -4.

- Horizontal Asymptote: Compare the degrees of the numerator and denominator.
Since the degree of the numerator is equal to the degree of the denominator, there is a horizontal asymptote at y = 1 (the ratio of the leading coefficients).

- x-Intercept: Set the numerator equal to zero and solve for x.
x - 2 = 0
x = 2
So, there is an x-intercept at (2, 0).

- y-Intercept: Set x = 0 and evaluate the function.
f(0) = (0 - 2) / (0 + 4) = -1/2
So, there is a y-intercept at (0, -1/2).

2. For f^r = (x + 4) / (x - 2):
- Vertical Asymptote: Set the denominator equal to zero and solve for x.
x - 2 = 0
x = 2
So, there is a vertical asymptote at x = 2.

- Horizontal Asymptote: Compare the degrees of the numerator and denominator.
Since the degree of the numerator is equal to the degree of the denominator, there is a horizontal asymptote at y = 1 (the ratio of the leading coefficients).

- x-Intercept: Set the numerator equal to zero and solve for x.
x + 4 = 0
x = -4
So, there is an x-intercept at (-4, 0).

- y-Intercept: Set x = 0 and evaluate the function.
f^r(0) = (0 + 4) / (0 - 2) = -2
So, there is a y-intercept at (0, -2).

b) Symmetry:
The graphs of both f(x) and f^r are symmetric with respect to the line y = x. This means that if you folded the graph along the line y = x, the two sides would overlap each other perfectly.

c) Intervals of Increase and Decrease:
To determine the intervals of increase and decrease, we need to examine the behavior of the functions between critical points (where the derivative is zero or undefined) and at the asymptotes.

For f(x) = (x - 2) / (x + 4):
- f is increasing to the left of -4 (vertical asymptote) and decreasing to the right of -4.
- f has a minimum point at x = 2 (x-intercept).

For f^r = (x + 4) / (x - 2):
- f^r is increasing to the left of 2 (vertical asymptote) and decreasing to the right of 2.
- f^r has a minimum point at x = -4 (x-intercept).

The sets of intervals for f and f^r are the same. They are both increasing to the left of their respective vertical asymptotes and decreasing to the right.

d) Pattern for Other Functions:
No, the pattern would not occur for all pairs of functions f(x) = x + b (numerator) / (x + d) and f^r(x) = x + d (numerator) / (x + b) because the placement of the asymptotes and intercepts would change based on the values of b and d. The symmetry may not be present, and the intervals of increase and decrease can differ. The specific values of b and d would determine the behavior of the functions.