Let's analyze the function $F(x) = \frac{1}{(x - 9)^2}$ step by step to answer the questions.

(a) Graph the rational function using transformations

We start with the basic function $f(x) = \frac{1}{x^2}$, which has the following characteristics:

• It's always positive for all $x \neq 0$, with a vertical asymptote at $x = 0$.
• As $x \to 0$, the function approaches $+\infty$.
• As $x \to \pm \infty$, the function approaches $0$ (horizontal asymptote at $y = 0$).

Transformations applied to $f(x) = \frac{1}{x^2}$:

1. Horizontal shift: $F(x) = \frac{1}{(x - 9)^2}$ shifts the graph of $\frac{1}{x^2}$ 9 units to the right. So, the vertical asymptote moves to $x = 9$.

This gives us the graph of $F(x)$. It retains the same shape as $f(x) = \frac{1}{x^2}$, but it is centered at $x = 9$.

(b) Use the final graph to find the domain and range

Domain:

The domain of $F(x) = \frac{1}{(x - 9)^2}$ is all real numbers except where the denominator is zero. The denominator is zero when $x = 9$, so the function is undefined there. Thus, the domain is: $\boxed{(-\infty, 9) \cup (9, +\infty)}$

Range:

Since $(x - 9)^2$ is always positive for $x \neq 9$, and the function $\frac{1}{(x - 9)^2}$ outputs positive values, the graph of $F(x)$ stays above the x-axis, approaching zero but never reaching it. Thus, the range is: $\boxed{(0, +\infty)}$

(c) Asymptotes

Vertical Asymptote:

• There is a vertical asymptote at $x = 9$, as the function tends to infinity as $x \to 9$.

Horizontal Asymptote:

• As $x \to \pm \infty$, $F(x) = \frac{1}{(x - 9)^2} \to 0$, so there is a horizontal asymptote at $y = 0$.

Oblique Asymptote:

• There are no oblique asymptotes because the degree of the numerator is less than the degree of the denominator.

Summary:

1. Domain: $(-\infty, 9) \cup (9, +\infty)$
2. Range: $(0, +\infty)$
3. Vertical Asymptote: $x = 9$
4. Horizontal Asymptote: $y = 0$
5. No oblique asymptote

Would you like a detailed graph of the function?

---

Here are 5 questions to explore:

1. What is the behavior of the function near $x = 9$?
2. How does the transformation affect the graph's symmetry?
3. Can you determine the limits as $x \to 9^-$ and $x \to 9^+$?
4. What happens if we change the power in the denominator from 2 to 1?
5. How would a vertical shift affect the graph?

Tip: Asymptotes give critical insights into the long-term behavior of a function! Always check for them when dealing with rational functions.