3.3E   Techniques in Graphing

Translations  

Vertical Shift- Highlights  

~   The translation will be vertically f(x) + k. 

~    It will either move up or down the y-axis depending on the value of k. 

~    If k > 0, the graph will shift upward k units. 

~    If k < 0, the graph will shift downward k units. 

~    In this transformation you are adding the value of k to all of the function’s y-values. 

~     The function’s y-values change, which will result in the graph shifting up/down the y-axis.

Horizontal Shift - Highlights 

~  The translation will be horizontally f(x + k) 

~   It will either move left or right depending on the value of k 

~   If k > 0, the graph will shift to the left k units          

~   If k < 0, the graph will shift to the right k units.          

~   In this transformation, you are adding the value of k to all of the function’s x-values. 

~   The function’s x-values change, which will result in the graph’s shifting left/right along the x-axis.

Reflections 

Vertical Reflection - Highlights 

~   The transformation will be a reflection over the x-axis.  ~    -f(x) reflects f(x) over the x-axis.          

~    Think of this transformation as a mirror reflection (or folding the graph over the x-axis). 

~   The reflection will affect the y-coordinate -> the y-values will become negative. 

~     If you have (x, y) the reflection would be (x, -y).

Horizontal Reflection  - Highlights 

~ The transformation will be a reflection over the y-axis. 

~  f(-x) reflects f(x) over the y-axis. 

~ Think of this transformation as a mirror reflection (or folding the graph over the y-axis. 

~ The reflection will affect the x-coordinates -> the x-values will become negative. 

~  If you have (x, y), the reflection  would be (-x, y)

Dilations

Dilations Vertical Stretch  - Highlights 

~   kf(x) -  The transformation will stretch/shrink f(x) vertically 

~   If k > 1, the graph of y = k f(x) will be stretched vertically. 

~   The function’s vertical stretching will result in its graph stretching away fr om the x-axis. 

~    Multiply each y-coordinate by k. 

Vertical Compression (or shrinking) - Highlights          

~    If 0 < k < 1 (which is a fraction), then y = k f(x) will be squeezing toward the x-axis. 

~    Multiply each y-coordinate by k.

Dilations Horizontal Stretch  - Highlights 

~   f(kx)  - The transformation will stretch/shrink f(x) horizontally.

 ~   If 0 < k < 1 (a fraction), the graph of f(kx) will be stretched horizontally. 

~   The function’s horizontal stretching will result in its graph stretching away from the y-axis. 

~    Divide each x-coordinate by k. 

Horizontal Compression (or shrinking)  - Highlights 

~    If k > 1, then y = f(kx) will be s