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.
- 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?
- 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?
- Again re-center the function, and repeat questions 1 & 2 for the function y=| x |.
- 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?
- Repeat question 4 for the functions y=x² and y=| x |.
- 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.
- 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?
- 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.
- 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?
- How do you expect the graph of y=(0.5 x)² to be related to the graph of y=x²? Try it.