Below is a graph of [math]y=2^x[/math] which is a green curve and [math]y=a\times2^x+b[/math] which is a blue curve, which may be on top of the green curve if [math]a=1[/math] and [math]b=0[/math].[br][br]We are going to change the values of [math]a[/math] and [math]b[/math] and consider the changes to the shape and position of the curve.

Increase the value of [math]b[/math]. What happens to the curve?[br][br]Decrease the value of [math]b[/math]. What happens to the curve?[br][br]Tick the "Show Asymptote" check box. The asymptote should be shown as a dashed orange line.[br][br]How does the equation of the asymptote change as the value of [math]b[/math] changes. (remember that horizontal lines have the form [math]y=number[/math].

Return [math]b[/math] to zero. With [math]a=1[/math] the blue curve should overlap the green curve.[br][br]Select the "Show Intercept" check box.[br][br]Change to [math]a=2[/math]. What is the intercept now? How far above the asymptote is the intercept?[br][br]Change to [math]a=3[/math]. What is the intercept now? How far above the asymptote is the intercept?[br][br]Do the same with [math]a=4[/math] and [math]a=5[/math]. What do you notice?

What happens when [math]a=-1[/math]? How does the position of the intercept compare to the asymptote?[br][br]Try changing to [math]a=-2[/math], [math]a=-3[/math], [math]a=-4[/math], and [math]a=-5[/math]. Do you see the same thing with regard to the asymptote and the intercept?

With [math]a=1[/math], change to [math]b=1[/math].[br][br]Try to predict what is going to happen when you now change [math]a[/math]. Do you expect to see the same relationship between the position of the asymptote and the position of the intercept that we saw before?[br][br]Change the value of [math]a[/math]. Do you see what you predicted?[br][br]Try various values of [math]a[/math] and [math]b[/math]. Do you see what you expect.