Question

1. find the vertical and horizontal asymptotes:

a. (f(x) = \frac{1}{x+2} + 3)
b. (f(x) = \frac{1}{x-2} - 3)
c. (f(x) = \frac{1}{x-4} + 2)

1. write a rational function that has a vertical asymptote at (x = 7) and a hole at (x = -4).

Explanation:

Analyze the orientation of the image

The image is upside down. Let's read the text by rotating it 180 degrees:

• Question 6: Find the vertical and horizontal asymptotes:
• a. \(f(x) = \frac{1}{x+2} + 3\)
• b. \(f(x) = \frac{1}{x-2} - 3\)
• c. \(f(x) = \frac{1}{x-4} + 2\)
• Question 7: Write a rational function that has a vertical asymptote at \(x = 7\) and a hole at \(x = -4\).

Determine asymptotes for Question 6

Using the Rational Graphs knowledge point
For a rational function in the form \(f(x) = \frac{a}{x-h} + k\):

• The vertical asymptote occurs where the denominator is zero: \(x - h = 0 \implies x = h\).
• The horizontal asymptote is the horizontal shift: \(y = k\).

Let's apply this to each sub-question:

• 6a: \(f(x) = \frac{1}{x+2} + 3\)
• Vertical asymptote: \(x + 2 = 0 \implies x = -2\)
• Horizontal asymptote: \(y = 3\)
• 6b: \(f(x) = \frac{1}{x-2} - 3\)
• Vertical asymptote: \(x - 2 = 0 \implies x = 2\)
• Horizontal asymptote: \(y = -3\)
• 6c: \(f(x) = \frac{1}{x-4} + 2\)
• Vertical asymptote: \(x - 4 = 0 \implies x = 4\)
• Horizontal asymptote: \(y = 2\)

Construct the rational function for Question 7

Using the Rational Expressions knowledge point
To construct a rational function \(f(x) = \frac{P(x)}{Q(x)}\):

• A vertical asymptote at \(x = 7\) means there is a factor of \((x - 7)\) in the denominator that does not cancel out.
• A hole at \(x = -4\) means there is a common factor of \((x - (-4)) = (x + 4)\) in both the numerator and the denominator.

Thus, we can write the function as:

$$f(x) = \frac{x + 4}{(x - 7)(x + 4)}$$

Answer:

Question 6

a. Vertical asymptote: \(x = -2\), Horizontal asymptote: \(y = 3\)
b. Vertical asymptote: \(x = 2\), Horizontal asymptote: \(y = -3\)
c. Vertical asymptote: \(x = 4\), Horizontal asymptote: \(y = 2\)

Question 7

An example of such a rational function is:

$$f(x) = \frac{x + 4}{(x - 7)(x + 4)}$$