Question

rational expressions and functions
student activity sheet 3; exploring \modeling with rational functions\
page 4 of 9

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 \(y = \frac{1}{x}\). The graph of \(y=\frac{1}{x}\) has two branches in the first and third quadrants.

Step2: Analyze the transformation

For the function \(g(x)=\frac{-0.5}{x}=-0.5\times\frac{1}{x}\), there are two transformations:

• A vertical compression by a factor of \(0.5\). For a function \(y = a\times\frac{1}{x}\), when \(|a|\lt1\) (\(a = 0.5\) here), the graph of \(y=\frac{1}{x}\) is vertically compressed. If \((x,y)\) is on the graph of \(y=\frac{1}{x}\), then \((x,0.5y)\) is on the graph of \(y = 0.5\times\frac{1}{x}\).
• A reflection about the \(x\) - axis. Since \(a=-0.5\), the graph of \(y = 0.5\times\frac{1}{x}\) is reflected about the \(x\) - axis. If \((x,y)\) is on the graph of \(y = 0.5\times\frac{1}{x}\), then \((x,-y)\) is on the graph of \(y=-0.5\times\frac{1}{x}\).

To graph \(g(x)=\frac{-0.5}{x}\):

1. First, recall the key points of \(y = \frac{1}{x}\) such as \((1,1)\) and \((- 1,-1)\).
2. For the vertical compression by a factor of \(0.5\), the points \((1,1)\) and \((-1,-1)\) on \(y=\frac{1}{x}\) transform to \((1,0.5)\) and \((-1,-0.5)\) on \(y = 0.5\times\frac{1}{x}\).
3. Then, for the reflection about the \(x\) - axis, the points \((1,0.5)\) and \((-1,-0.5)\) transform to \((1,-0.5)\) and \((-1,0.5)\) on \(y=\frac{-0.5}{x}\).

The asymptotes of \(y=\frac{-0.5}{x}\) are still the \(x\) - axis (\(y = 0\)) and \(y\) - axis (\(x = 0\)) because the transformations (vertical compression and reflection) do not affect the asymptotes of the rational function \(y=\frac{1}{x}\).

Answer:

First, start with the parent function \(y=\frac{1}{x}\) (with asymptotes \(x = 0\) and \(y=0\)). Then, apply a vertical compression by a factor of \(0.5\) (so \(y = 0.5\times\frac{1}{x}\)) and a reflection about the \(x\) - axis (resulting in \(y=\frac{-0.5}{x}\)). Plot key points (adjusted for compression and reflection) and draw the two - branch hyperbola with asymptotes \(x = 0\) and \(y = 0\).