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B. Reciprocals of Quadratic Functions

 Investigation


Reciprocals of Quadratic Functions


Open the Reciprocal of a Function applet. The applet can show various linear and quadratic functions as well as their reciprocals.

Turn on “quadratic”. The graph of the quadratic function \(y = g(x)\) can be adjusted using the \(a\)-, \(h\)-, and \(k\)-values. Turn on “show reciprocal”. This shows the graph of \(y = \frac{1}{g(x)}\).

  1. How are the two graphs related?

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

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

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

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



    When \(y = f(x)\) represents a quadratic function, the graph of \(y = \frac{1}{f(x)}\) can take on several different shapes. However, the relationships between the graphs of \(y = f(x)\) and \(y = \frac{1}{f(x)}\) that you explored when \(y = f(x)\) was linear still hold.


 

 Key Lesson Marker



  • Vertical asymptotes of the graph of \(y = \frac{1}{f(x)}\) occur when \(f(x) = 0\).
  • There is a horizontal asymptote at \(y = 0\).
  • The graphs of \(y = f(x)\) and its corresponding reciprocal function \(y = \frac{1}{f(x)}\) intersect when the \(y\)-values are \(\pm 1\). These are the invariant points.


As with linear functions and their reciprocals, the graph of the reciprocal of a quadratic function can be sketched using the graph of the quadratic function.