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General Review of Function Transformation

General Function Transformations Many functions can be described as transforms of a parent function. We may perform several transformations on a parent function:
  • Vertical Stretch and Reflection
  • Horizontal Stretch and Reflection
  • Horizontal Shift (also called Horizontal Translation)
  • Vertical Shift (also called Vertical Translation)
Below we have the parent function which will allow us to investigate vertical stretch and reflection and the horizontal and vertical shifts. We will address horizontal stretch using a different function below. I. Vertical Stretch and Reflection In the applet, We see that the child function of the parent with vertical stretch takes the form

________________________________________________________________________________________ Experiment: Change the value of and see what happens to the child function. Note how governs both vertical stretch and reflection. Vertical reflection occurs when is a negative value.

State your observations regarding the changes in the shape of the quadratic function as changes value.

________________________________________________________________________________________ II. Horizontal Shift In the applet, We see that the child function with only vertical stretch and horizontal shift is Pay attention to how the child function is written.
  • The variable and the horizontal shift are grouped, , and then squared.
  • Horizontal shift is subtracted from
Example:
  • is a parabola that has been translated to the right 4 units.
  • is a parabola that has been translated to the left 3 units.
________________________________________________________________________________________ Experiment: Change the values of and whist taking note of the equation of the child function.
________________________________________________________________________________________ III. Vertical Shift In the applet, We see that the child function that has vertical stretch and both horizontal and vertical shift is



For a quadratic, you will notice that corresponds to the vertex of the parabola. ________________________________________________________________________________________ Experiment: Change the values of whist taking note of the equation of the child function and the behaviour of the function.

Describe what transformations have been performed on the parent function to yield

(Yes, algebraic manipulation is needed - remember completing the square?)

________________________________________________________________________________________ IV. Horizontal Stretch To visualise horizontal stretch, we will look at the function

All the transformations discussed previously apply, which we can see applied to the child function

In addition, we have horizontal stretch/compression and reflection given by ________________________________________________________________________________________ Experiment: Set the values for to 1 and to 0. Change the value of to see how the asymptote at x=0 does not change. Compare the behaviour of the function as changes to that as . They are different! The most striking difference occurs when these coefficients change sign.

Describe the difference that you see in the graph when you change horizontal stretch compared to vertical stretch for the transformation of the logarithmic function. What happens when either of these values turns negative?

________________________________________________________________________________________ V. Summary In this activity, we have reviewed the main transformation that may be performed on a parent function:
  • Vertical stretch and reflection,
  • Horizontal stretch and reflection,
  • Vertical shift, and
  • Horizontal shift
________________________________________________________________________________________

Explain how the examination of a function's graph allows you to discern which transformations a parent function has undergone to yield its child. Use examples during your discussion.