Question

1. reinforce use your knowledge of transformations of functions to graph the function ( h(x)=-\frac{5}{x} ). describe the strategy you used.

Explanation:

Step1: Identify the parent function

The parent function of \( h(x)=-\frac{5}{x} \) is \( y = \frac{1}{x} \). The graph of \( y=\frac{1}{x} \) has two branches in the first and third quadrants.

Step2: Consider the vertical stretch

For the function \( y = a\frac{1}{x}\), when \(|a|> 1\) (here \(a=-5\)), it is a vertical stretch. For \(y=\frac{1}{x}\), when \(x = 1\), \(y = 1\); for \(y=-5\frac{1}{x}\), when \(x = 1\), \(y=-5\).

Step3: Consider the reflection

The negative sign in \(y=-5\frac{1}{x}\) reflects the graph of \(y = 5\frac{1}{x}\) (after vertical - stretch) over the \(x\) - axis. The graph of \(y=\frac{1}{x}\) is in the first and third quadrants. After reflection over the \(x\) - axis, the graph of \(y =-\frac{1}{x}\) is in the second and fourth quadrants. After vertical stretch (\(a = 5\)), for \(h(x)=-\frac{5}{x}\), we can plot points:

• When \(x=-1\), \(h(-1)=5\)
• When \(x = - 0.5\), \(h(-0.5)=10\)
• When \(x=0.5\), \(h(0.5)=-10\)
• When \(x = 1\), \(h(1)=-5\)

Answer:

To graph \(h(x)=-\frac{5}{x}\), start with the parent function \(y = \frac{1}{x}\). First, perform a vertical stretch by a factor of \(5\) (multiply the \(y\) - values of \(y=\frac{1}{x}\) by \(5\)). Then, reflect the resulting graph (of \(y = \frac{5}{x}\)) over the \(x\) - axis. Plot key points such as \((-1,5)\), \((-0.5,10)\), \((0.5,-10)\), \((1,-5)\) and draw the two - branch hyperbola in the second and fourth quadrants.