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)
State your observations regarding the changes in the shape of the quadratic function as changes value.
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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
- is a parabola that has been translated to the right 4 units.
- is a parabola that has been translated to the left 3 units.
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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?)
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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
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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?
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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.