Question

chapter iv.1
score: 1/5 answered: 2/5
question 3
take $f$ to be the function that is given by
$f(x)=(x + 9)|x + 9|-6$.
determine a point $p$ so that the rotation of $f$ around $p$ by half of a circle is equal to $f$.

Explanation:

Step1: Analyze the function's symmetry

The function \(y = f(x)=(x + 9)|x + 9|-6\) is a transformation of the function \(y = x|x|\). The function \(y=x|x|=

$$\begin{cases}x^{2},&x\geq0\\-x^{2},&x<0\end{cases}$$

\) is symmetric about the origin \((0,0)\). The function \(y=(x + 9)|x + 9|-6\) is a translation of \(y = x|x|\) 9 units to the left and 6 units down.

Step2: Determine the center of symmetry

For a function \(y = g(x - a)+b\) which is a translation of \(y = g(x)\) by \(a\) units to the right and \(b\) units up, if \(g(x)\) is symmetric about the origin, then \(y = g(x - a)+b\) is symmetric about the point \((a,b)\). Here \(a=- 9\) and \(b = - 6\). A rotation of a function by half - a circle (180 degrees) about its center of symmetry results in the same function.

Answer:

\((-9,-6)\)