Glossary Checklist FAQ
 Investigation

Reciprocal Functions


  1. The function \(w = l\) can be used to represent a square (width equals length), and the function \(w = \frac{1}{l}\) can be used to represent a rectangle with an area of \(1\).

    1. What happens to \(w\) as \(l\) increases in each function?

    2. Predict the shape of the graph of each function.

    3. Complete the following tables of values, and plot the points to check your predictions from part b.

      \(w = l\)
      \(l\) \(w\)
      \[\frac{1}{4}\]
      \[\frac{1}{3}\]
      \[\frac{1}{2}\]
      \(1\)
      \(2\)
      \(3\)
      \(4\)  
      \(w = \frac{1}{l}\)
      \(l\) \(w\)
      \[\frac{1}{4}\]
      \[\frac{1}{3}\]
      \[\frac{1}{2}\]
      \(1\)
      \(2\)
      \(3\)
      \(4\)  


    4. How are the \(w\)-values related when the \(l\)-value is the same for each function?



    5. For which \(l\)-value(s) is there no relationship between the two functions' \(w\)-values?

  2. Open the Reciprocal of a Function applet. The applet can show various linear and quadratic functions, as well as their reciprocals. You’ll return to use the quadratic part of this applet later.

    Turn on “linear”. The graph of the linear function \(y = f(x)\) can be adjusted using the \(m\)- and \(b\)-values. Turn on “show reciprocal”. This shows the graph of \(y = \frac{1}{f(x)}\).

    1. How are the two graphs related?

    2. Turn on “show vertical asymptotes”. What do asymptotes represent on the graph? Can you predict the location of an asymptote using only \(y = f(x)\)? Only \(y = \frac{1}{f(x)}\)?

    3. Turn off “show vertical asymptotes” and turn on “show \(y = \pm 1\)”. Explain why the graphs of the two functions always intersect at either \(y = 1\) or \(y = -1\).

    4. For any \(x\)-value, if the \(y\)-values for \(y = f(x)\) and \(y = \frac{1}{f(x)}\) are not equal, the graphs of the two functions lie on opposite sides of \(y = 1\) or on opposite sides of \(y = -1\). Why is this?

    5. The graph of \(y = h(x)\) is shown on the grid. On the same grid, graph \(y = 1\), \(y = -1\), and the location of any asymptotes of the graph of \(y = \frac{1}{h(x)}\). Use these lines to sketch \(y = \frac{1}{f(x)}\).