1. Graphing Techniques: Transformations Transformations: Review
We will be looking at functions from our library of functions and seeing how various modifications to the functions transform them.
Transformations
Transformations
Transformations
Transformations
Transformations

2. VERTICAL TRANSLATIONS
Above is the graph of
As you can see, a number added or subtracted from a function will cause a vertical shift or translation in the function.
VERTICAL TRANSLATIONS
What if we wanted to shift the graph down by 3? Notation: f(x) – 3 (This would mean taking all the function values and subtracting 3 from them).
What if we wanted to shift the graph up by 1? Notation: f(x) + 1 (This would mean taking all the function values and adding 1 to them).

3. VERTICAL TRANSLATIONS
So the graph f(x) + k, where k is any real number is the graph of f(x) but vertically shifted by k. If k is positive it will shift up. If k is negative it will shift down
VERTICAL TRANSLATIONS
Above is the graph of
What would f(x) + 2 look like?
What would f(x) - 4 look like?

4. HORIZONTAL TRANSLATIONS
Above is the graph of
As you can see, a number added or subtracted from the x will cause a horizontal shift or translation in the function but opposite way of the sign of the number.
What if we wanted to shift the graph 1 to the right? Notation: f(x-1) (This would mean taking all the x values and subtracting 1 from them before putting them in the function).
What if we wanted to shift the graph 2 to the left? Notation: f(x+2) (This would mean taking all the x values and adding 2 to them before putting them in the function).

5. HORIZONTAL TRANSLATIONS
So the graph f(x-h), where h is any real number is the graph of f(x) but horizontally shifted by h. Notice the negative. (If you set the stuff in parenthesis = 0 & solve it will tell you how to shift along x axis).
So the graph f(x-h), where h is any real number is the graph of f(x) but horizontally shifted by h. Notice the negative. (If you set the stuff in parenthesis = 0 & solve it will tell you how to shift along x axis).
shift right 3
Above is the graph of
What would f(x+1) look like?
So shift along the x-axis by 3
What would f(x-3) look like?

6. We could have a function that is transformed or translated both vertically AND horizontally.
up 3
left 2
Above is the graph of
What would the graph of look like?

7. and
If we multiply a function by a non-zero real number it has the affect of either stretching or compressing the function because it causes the function value (the y value) to be multiplied by that number.
DILATION
Let's try some functions from our library of functions multiplied by non-zero real numbers to see this.

8. Notice for any x on the graph, the new (red) graph has a y value that is 2 times as much as the original (blue) graph's y value.
Notice for any x on the graph, the new (green) graph has a y value that is 4 times as much as the original (blue) graph's y value.
So the graph a f(x), where a is any real number GREATER THAN 1, is the graph of f(x) but vertically stretched or dilated by a factor of a.
Above is the graph of
What would 2f(x) look like?
What would 4f(x) look like?

9. What if the value of a was positive but less than 1?
So the graph a f(x), where a is 0 < a < 1, is the graph of f(x) but vertically compressed or dilated by a factor of a.
Notice for any x on the graph, the new (green) graph has a y value that is 1/4 as much as the original (blue) graph's y value.
Notice for any x on the graph, the new (red) graph has a y value that is 1/2 as much as the original (blue) graph's y value.
Above is the graph of
What would 1/2 f(x) look like?
What would 1/4 f(x) look like?

10. What if the value of a was negative?
So the graph - f(x) is a reflection about the x-axis of the graph of f(x). (The new graph is obtained by "flipping“ or reflecting the function over the x-axis)
Notice any x on the new (red) graph has a y value that is the negative of the original (blue) graph's y value.
Above is the graph of
What would - f(x) look like?

11. There is one last transformation we want to look at.
So the graph f(-x) is a reflection about the y-axis of the graph of f(x). (The new graph is obtained by "flipping“ or reflecting the function over the y-axis)
Notice any x on the new (red) graph has an x value that is the negative of the original (blue) graph's x value.
Above is the graph of
What would f(-x) look like? (This means we are going to take the negative of x before putting in the function)

12. Summary of Transformations So Far
Do reflections BEFORE vertical and horizontal translations
If a > 1, then vertical dilation or stretch by a factor of a
If 0 < a < 1, then vertical dilation or compression by a factor of a
If a < 0, then reflection about the x-axis (as well as being dilated by a factor of a)
vertical translation of k
f(-x) reflection about y-axis
horizontal translation of h (opposite sign of number with the x)

14. reflects about the x -axis
We know what the graph would look like if it was from our library of functions.
moves up 1
Graph using transformations
reflects about the x -axis
moves right 2

15. There is one more Transformation we need to know.
Do reflections BEFORE vertical and horizontal translations
If a > 1, then vertical dilation or stretch by a factor of a
If 0 < a < 1, then vertical dilation or compression by a factor of a
If a < 0, then reflection about the x-axis (as well as being dilated by a factor of a)
vertical translation of k
f(-x) reflection about y-axis
horizontal translation of h (opposite sign of number with the x)
horizontal dilation by a factor of b

16. Vertical Dilation
Now complete the Changes of Scale and Origin for Graphs Booklet to explore this idea further and to consolidate all graphical transformations.

17. Acknowledgement
I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint.
Shawna has kindly given permission for this resource to be downloaded from and for it to be modified to suit the Western Australian Mathematics Curriculum.
Stephen Corcoran
Head of Mathematics
St Stephen’s School – Carramar