Click on the picture to see an animation of this shift.  QT 

Can you anticipate what the graph of

would look like?  Click here to see an animation of what this shift would be.  QT

Click on the picture to see an animation of this shift.  QT

Can you anticipate what the graph of

would look like?  Click here to see an animation of what this shift would be.  QT

Click on the picture to see an animation of these shifts.  QT 

Can you anticipate what the graph of

would look like?  Click here to see an animation of what these shifts would be.  QT

Click on the picture to see an animation of this stretch.  QT

Click on the picture to see an animation of this "squash".  QT

where k varies from 1 to -1?  What is the relationship between the two graphs when k = -1?

The picture at the right shows the relationship when k = -1.  Observe that when k = -1 the graph of g is a reflection of the graph of f about the x-axis.  Click on the picture to see an animation with k going from 1 to -1.  QT

Click on the picture to see a larger picture.

f  g  h  p  q  r

if h and k are both positive?  What would be the relationship if h and k are both negative?  Could you write general rules that would cover all possibilities for h and k?  Hint:  Try using absolute value in the statement of your rules.

Could you write general rules that would cover all possibilities for k in comparing

Click here to see a DPGraph representation of y = x² and y = c(x - a)² + b.

The initial values for a, b, and c are 0, 0, and 1.  You can use the scrollbar to change a, b, and c and observe the corresponding shift (in the case of a and b) and stretch or compress (squash) in the case of c > 0.  You can also observe the effect of a negative c value.

Click here to see a DPGraph representation of y = x³ and y = c(x - a)³ + b and use the scrollbar to change a, b, and c.

In trigonometry the function f(x) = sin(x) (called the sine function) is studied.  Many of you will not have studied trigonometry at this point in your career in mathematics but the principles being introduced here will apply equally well to the sine function.

Click here to see a DPGraph representation of y = sin(x) and y = c[sin(x - a)] + b and use the scrollbar to change a, b, and c.

If you have studied trigonometry you might be interested in this final example.

Click here to see a DPGraph representation of y = sin(x) and y = c[sin(dx - a)] + b and use the scrollbar to change a, b, c, and d.