1. Unit #1 Transformations
1.1 Functions and Graphs
Functions vs. Relations
Relations An algebraic relationship between two variables (usually x and y)
Functions A relation illustrating one-to-one or many-to-one mapping (think vertical line test)

2. 1.1 Functions and Graphs cont.
Review of Pure Math 20 Functions
Domain
Range
x-intercept
y-intercept
none

3. 1.1 Functions and Graphs cont.
Pure Math 20 Function Skills
Finding the x-intercepts
Finding the y-intercepts
Find the x-intercepts of the function
Find the y-intercepts of the function
Workbook pg 3
Workbook pg 3

4. 1.2 Translating Functions
Translating – movement left/right (horizontal) or up/down (vertical)
Horizontal Translation
Examine the three functions shown below, and the resulting graphs based on
(The function f is simply cubing a value (x)
Examine the changes;

5. 1.2 Translating Functions cont.
Vertical Translation
Examine the three functions shown below, and the resulting graphs based on
Examine the changes;

6. 1.2 Translating Functions cont.
Translation Conclusions
is a translation of
h units to the left
is a translation of
h units to the right
is a translation of
k units down
is a translation of
k units up

7. 1.2 Translating Functions cont.
To graph any of the functions, create a stat plot, and choose correct lists for x and y.
EXAMPLE #1
Class example #4, page 16
Given
sketch
Graphing
Shift original function 2 units to the right
Repeat this process to create the third graph by moving the original graph 3 units left and 2 units down.
Enter y-values in a list
Enter x-values in a list
Set appropriate window
Graphing

8. 1.2 Translating Functions cont.
EXAMPLE #2
The following translations are based on the function
Describe how the function
relates to
The new function is a translation 2 units up and 1 unit left If
represent a function in terms of f which is translated 1 unit left.

9. 1.3 Reflecting Functions
There are three categories of reflections to examine.
Reflections around a horizontal line (often the x-axis)
Reflections around a vertical line (often the y-axis)
Reflections around the line y = x (called the inverse from Pure Math 20)

10. 1.3 Reflecting Functions cont.
Reflections around a horizontal line
Beginning with the function
examine the transformation
Consider
and describe the transformation
Conclusion
Is a reflection about the y-axis with all x coordinates multiplied by -1.

11. 1.3 Reflecting Functions cont.
Reflections around a vertical line
Beginning with the function
examine the transformation
Consider
and describe the transformation or
Conclusion
Is a reflection about the x-axis with all y coordinates multiplied by -1.

12. 1.3 Reflecting Functions cont.
The special case of inverse
Beginning with the function
examine a transformation which reflects the function about the line y = x.
Conclusion
Is a reflection about the line y = x. All elements of each ordered pair are interchanged. Thus, the ordered pair (a, b) become (b, a)

13. 1.3 Reflecting Functions cont.
Reflection Conclusions
A horizontal reflection about the y-axis with all x coordinates multiplied by -1.
A vertical reflection about the x-axis with all y coordinates multiplied by -1.
A reflection about the line y = x, with all ordered pairs interchanged.
A specific type of inverse reflection where the result of the inverse is also a function.

14. 1.3 Reflecting Functions cont.
Invariant Points
Any point(s) on an image, that when transformed, remain unchanged are called invariant point(s).
Examine the two functions shown below, where one is a transformation of the other. Notice, the transformed function can be described by two type of transformations.
If the transformed function is a translation 4 units to the right, there are no invariant points.
If the transformed function is a reflection about the y-axis, (0,1) is an invariant point. When (0,1) has it’s x-value multiplied by -1, the result is unchanged.

15. 1.3 Reflecting Functions cont.
EXAMPLE #1
Class example #1, pg. 29.
Enter all x-coordinates in L1 and all y-coordinates in L2
Create a STAT-PLOT with the appropriate window settings.
Multiply all L2 by -1, and store new values in L3. Re-plot using L1 (x) and L3 (y)
Multiply all L1 by -1, and store new values in L4. Re-plot using L4 (x) and L2 (y)
The x and y values must be interchanged. Simply re-plot using L2 (x) and L1 (y).

16. 1.3 Reflecting Functions cont.
EXAMPLE #2
Class example #4, pg. 34

17. 1.4 Stretching Functions
When examining expansions and compressions of functions, we call both a stretch. Stretches can occur about any line, although we will examine stretches about the x-axis, y-axis, and line parallel to either axis.
Stretches about the y-axis
As in a reflection about the y-axis caused by a change in x, the same holds true for a stretch about the y-axis.
Examine the function
and its transformation
We can express this as
Conclusion
Is a stretch by a factor of ½ about the y-axis.
Must be a stretch by a factor of 2 about the y-axis.

18. 1.4 Stretching Functions cont.
Stretches about the x-axis
As in a reflection about the x-axis caused by a change in y, the same holds true for a stretch about the x-axis.
Examine the function
and its transformation
We can express this as
Conclusion
Is a stretch by a factor of ½ about the x-axis.
Must be a stretch by a factor of 2 about the x-axis.

19. 1.4 Stretching Functions cont.
Stretches about any horizontal or vertical line
The function
is stretched by a factor of 2 about the line
What does this look like, and how can it be done? We first need to know some information.
What is the function we are stretching?
Where is the line about which we are stretching?
What is the factor of the stretch?
Notice the creation of three points, A, B and C. Each of these points are twice the distance from the line of stretch (factor of 2) relative to their original location. Points on the line are invariant.
And now, we can create the transformed image.

20. 1.4 Stretching Functions cont.
Stretching Conclusions
A horizontal stretch by a factor of 1/k about the y-axis where all x values are multiplied by 1/k.
A horizontal stretch by a factor of k about the y-axis where all x values are multiplied by k.
A vertical stretch by a factor of 1/k about the x-axis where all y values are multiplied by 1/k.
A vertical stretch by a factor of k about the x-axis where all y values are multiplied by k.
When stretching an object about a fixed horizontal or vertical line, choose key point to re-plot. Determine the distance those point lie from the line about which the stretch occurs, and apply the appropriate stretch factor. Pay attention to invariant points.

21. 1.4 Stretching Functions cont.
EXAMPLE #1
Class example #1, pg. 44.
Write the replacement for x or y and write the equation of the image of y=f(x) after each transformation.
a. A stretch by a factor of 6 about the y-axis.
b. A stretch by a factor of 1/5 about the x-axis.
c. A reflection and stretch by a factor of 3, both in the x-axis.
d. A stretch about the y-axis by ½ and a stretch about the x-axis by ¼.

22. 1.4 Stretching Functions cont.
EXAMPLE #2
Class example #4, pg. 45.
This is a reflection about the y-axis and a stretch by a factor of ½, also about the y-axis.
Enter all x-values in L1 and y-values in L2.
Create a STAT-PLOT using the appropriate window.
Both changes are horizontal, so multiply all x-values (L1) by -1/2 and store in L3
Re-plot using L3 (x) and L2 (y).

23. 1.5 Combination Transformations
So far, we have examined transformations (translations, reflections and stretches) each of which have occurred one at a time.
Is it possible for these to occur in combination? Yes. But, we must pay attention to the following;
what are the actual transformations that occur and,
what order to they occur in?
Let’s build a series of transformations to examine how the order of the transformations might be important.
We’ll start with
and define it as
Reflect the function about the x-axis
Stretch about the x axis by a factor of 1/2
Stretch about the y-axis by a factor of 3.
Translate 4 units to the left.

24. 1.5 Combination Transformations cont.
Here is the rule to follow, particularly when you are looking at a set of horizontal transformations or a set of vertical transformation, and you must decide the order they occur.
If the stretch is factored from the translation, perform the stretch first, then the translation (read left to right).
If the stretch is blended with the translation (expanded through multiplication), then perform the translation before the stretch.
It will be to your advantage to factor any stretches before analyzing the result.

25. 1.5 Combination Transformations cont.
Combination Transformation Examples
Stretch factor of 1/3. This is factored out, so we read left to right.
Stretch factor of 4. This is not factored out, so we read right to left.
Translate 1 unit right.
Stretch about the y-axis by 4
Stretch about the y-axis by 1/3
Translate 2 units left

26. 1.5 Combination Transformations cont.
Recognizing Similar Transformations
Function #1
Function #2
Conclusion
Although the order of transformation is different, the result is the same.
The order of the transformations are different, and the result based on those transformations is different.
The order of the transformations are different, and the result based on those transformations is different. The rule does apply to vertical changes as well.

27. 1.5 Combination Transformations cont.
Recognizing two different transformations with the same result
Examine the following transformation, applied to the function
Stretch by ½ about the y-axis
Translate 1 unit right.
Translate 2 units right
Stretch ½ about the y-axis
Although the order is different, the result is the same.

28. 1.5 Combination Transformations cont.
The best way to handle combination transformations
FACTOR THE STRETCH!
Factor and use
Factor and use
Factor and use

29. 1.5 Combination Transformations cont.
EXAMPLE #1
Class example #4, pg. 69
First, enter all x-values in L1 and y-values in L2. Plot the function.
Re-arrange the expression to ensure easier ordering.
Decide the order you will perform the transformations. In this case;
Vertically translate 8 units down (L2-8) and store in L4.
Stretch by a factor of 2 about the y-axis and translate 6 units left (2*L1-6) and store in L3
Re-plot using L3 (x) and L4 (y).

30. 1.6 Reciprocal Transformations
From Pure Math 20
The transformation we must examine is
This is NOT the same as the inverse
Here are the steps, as reviewed from Pure Math 20;
Locate all x-intercepts on f(x). These are the locations of your asymptotes.
Locate all ordered pairs on f(x) which contain a y-value of 1 or -1. These are your invariant points.
As the function f(x) approaches the x-axis, its reciprocal moves away from the x-axis. As the function f(x) moves away from the x-axis, its reciprocal moves away from the x-axis.

31. 1.6 Reciprocal Transformations cont.
EXAMPLE #1
Class example #2 (ii), pg. 88
Locate the asymptotes.
Locate any invariant points. In this figure, they are represented by A, B, C and D.
Sketch the reciprocal, keeping in mind the function must stop at x = -4 and 5.

32. 1.6 Reciprocal Transformations cont.
EXAMPLE #2
Class example #3 (ii), pg. 89
a. Information from g(x) will help to find f(x). Specifically,
Minimum:
y-intercept
b. Use y = x2 as the base equation, and determine the solution.