Question

1. reinforce use your knowledge of transformations of functions to graph the function

$g(x)=\frac{-0.5}{x}$. describe the strategy you used.

Explanation:

Step1: Identify the parent function

The parent function of \(g(x)=\frac{- 0.5}{x}\) is \(y = \frac{1}{x}\).

Step2: Analyze the transformation

• The negative sign in front of \(0.5\) reflects the graph of \(y=\frac{1}{x}\) over the \(x\) - axis.
• The coefficient \(0.5\) (or \(\frac{1}{2}\)) vertically compresses the graph of \(y = \frac{1}{x}\) by a factor of \(\frac{1}{2}\).

To graph \(y=\frac{1}{x}\), we can find some key points. For example, when \(x = 1\), \(y = 1\); when \(x=-1\), \(y=-1\).

After reflection over the \(x\) - axis, the point \((1,1)\) becomes \((1, - 1)\) and \((-1,-1)\) becomes \((-1,1)\).

After vertical compression by a factor of \(\frac{1}{2}\), the point \((1,-1)\) becomes \((1,-0.5)\) and \((-1,1)\) becomes \((-1,0.5)\)

We also know that the function \(y=\frac{k}{x}\) has two branches. The \(x\) - axis (\(y = 0\)) is the horizontal asymptote and the \(y\) - axis (\(x = 0\)) is the vertical asymptote.

Answer:

First, start with the parent function \(y=\frac{1}{x}\). Reflect it over the \(x\) - axis (due to the negative sign) and then vertically compress it by a factor of \(0.5\). Plot key points such as \((1,-0.5)\), \((-1,0.5)\) and use the asymptotes \(x = 0\) and \(y = 0\) to sketch the two - branch hyperbola of the function \(g(x)=\frac{-0.5}{x}\)