For some of the following functions, work out the values of \(f(1)\), \(f(-1)\), \(f(2)\), \(f(-2)\), and so on.

  • \(f(x)=x^2\)

  • \(f(x)=2\)

  • \(f(x)=1-x^2\)

  • \(f(x)=x^4+2\)

  • \(f(x)=1-\dfrac{1}{x^2}\)

  • \(f(x)=x^2-2x+2\)

What did you notice as you worked out these values?

Now carefully sketch the graphs of the functions on the same set of axes.

  • What did you notice as you sketched the graphs?

  • Did your ideas change as you added new graphs to your axes?

  • What are the points of intersection of some of the graphs?

On a new set of axes, carefully sketch graphs of the following functions. Again, for some of the functions you may find it helpful to work out \(f(1)\), \(f(-1)\), \(f(2)\), \(f(-2)\), and so on. Note down your ideas as you sketch each graph.

  • \(f(x)=x\)

  • \(f(x)=-\dfrac{1}{x}\)

  • \(f(x)=x^3\)

  • \(f(x)=\dfrac{x^3}{10}\)

  • \(f(x)=x(x-1)(x+1)\)

  • \(f(x)=1+\dfrac{1}{x}\)

What did you notice this time?