Question

5 f(x)=(3x - 1)(3x + 1)
dilate the function by x - 2.

Explanation:

Step1: Expand the original function

First, expand \(f(x)=(3x - 1)(3x + 1)\) using the difference - of - squares formula \((a - b)(a + b)=a^{2}-b^{2}\). Here \(a = 3x\) and \(b = 1\), so \(f(x)=9x^{2}-1\).

Step2: Apply the dilation rule

To dilate the function \(y = f(x)\) by a factor of \(k\) in the \(x\) - direction, we replace \(x\) with \(\frac{x}{k}\). Here, it seems there is a mis - statement. If it is a horizontal dilation by a factor related to \(x-2\), we assume it means a horizontal dilation by a factor of \(\frac{1}{1}\) (since no clear dilation factor in the standard sense is given in a proper way, and if we assume it's a non - standard way of saying a transformation related to \(x - 2\), and we consider the most likely case of horizontal shift and dilation). If we assume it's a horizontal shift of 2 units to the right (a more common interpretation when we see \(x-2\) in the context of function transformations), we replace \(x\) with \(x - 2\) in the function. So the new function \(g(x)=9(x - 2)^{2}-1\).
Expand \(9(x - 2)^{2}-1\):
\[

$$\begin{align*} 9(x - 2)^{2}-1&=9(x^{2}-4x + 4)-1\\ &=9x^{2}-36x+36 - 1\\ &=9x^{2}-36x + 35 \end{align*}$$

\]

Answer:

\(y = 9x^{2}-36x + 35\)