Prerequisites: function notation, graphing
Goal: To understand how a change in a function corresponds to a change in its graph.

To use this applet, you should know that there are two graphs: one in red and one in blue, with matching functions in red and blue below the graphs. At the beginning, only one function appears because the two are identical.
  1. Use the slider below the graph to shift the blue graph to the right by 1 unit. How does the function change? What do you expect the graphs of y=sin(x+1.6) and y=sin(x-0.5) will look like?
  2. Re-center the function. Change the function to y=x² by clicking on the top function in the column on the right, and then adjusting the exponent using the top slider. Again shift the blue graph to the right by 1 unit. How does the function change? What do you expect the graphs of y=(x+1.6)² and y=(x-0.5)² will look like?
  3. Again re-center the function, and repeat questions 1 & 2 for the function y=| x |.
  4. Re-center the graph, and again choose the function y=sin(x). Use the slider to the left of the graph to shift the blue graph up 1 unit. How does the function change? What do you expect the graphs of y=sin(x)+1.6 and y=sin(x)-0.5 will look like?
  5. Repeat question 4 for the functions y=x² and y=| x |.
  6. What do you expect the graph of y=sin(x+0.5)-2 to look like? Guess first, and use the applet to check your guess.
  7. If you want to produce a parabola whose vertex is shifted to the left by 0.5 and down by 0.3, what might the function be?
  8. Again considering y=sin(x), what happens to the function if we vertically stretch (using the second slider from the top on the right) the graph by a factor of 1.5? What do you expect the graph of y=(-0.5)sin(x) to look like? Guess first, then use the applet to check your guess.
  9. Predict the appearance of the graph of y=sin(2x). After writing your prediction, use the horizontal stretch slider (at top right) to produce the graph. Was your guess correct?
  10. How do you expect the graph of y=(0.5 x)² to be related to the graph of y=x²? Try it.