
Functions can be transformed (distorted) in various ways to help them better represent processes and behaviors found in the real world. You have already seen references to the concept of transformations in algebra and in geometry. The various types of functional transformations shown on this page will be a review, and enhancement, of those concepts.
Transformations are used to move, resize and distort graphs of functions.
We will be examining the following changes to f (x):
 f (x), f (x), f (x) + k, f (x + k), kf (x), f (kx)
reflections translations dilations
Reflections of Functions: f (x) and f (x) 
Vertical reflection:
reflection over the xaxis.
f (x) reflects f (x) over the xaxis


Reflections are mirror images.
Think of "folding" the graph over the xaxis.
On a grid, you used the formula
(x,y) → (x,y) for a reflection in the
xaxis, where the yvalues were negated.
In functional notation
we can write
(x, f (x)) → (x, f (x)).



Horizontal reflection:
reflection over the yaxis.
f (x) reflects f (x) over the yaxis 

Reflections are mirror images.
Think of "folding" the graph over the yaxis.
On a grid, you used the formula (x,y) → (x,y) for a reflection in the yaxis, where the xvalues were negated.
In function notation we can write
(x, f (x)) → (x, f (x)).

Translations of Functions: f (x) + k and f (x + k) 
Vertical shift:
vertical translation (straight up or straight down)
f (x) + k moves f (x) up or down

Changes occurs "outside" the function (affecting the yvalues). 
This translation is a "slide" straight up or down.
• if k > 0, the graph moves upward k units.
• if k < 0, the graph moves downward k units.
Remember the formula (x,y) → (x,y + k) moving a figure upward or downward on a graph.
In function notation, we can write
(x, f (x)) → (x, f (x) + k).
You are adding the value
of k to the yvalues of the function.



Horizontal shift:
horizontal translation (left or right)
f (x + k) moves f (x) left or right 
Changes occurs "inside" the function (affecting the xvalues). 
This translation is a "slide" left or right.
• if k > 0, the graph moves to the left k units.
• if k < 0, the graph moves to the right k units.
This can be very confusing!
k positive moves graph left
k negative moves graph right.
Consider this scenario: Suppose lunch = 11 am. If this changes to (lunch+2) = 11 am, then lunch is 9 am. Adding 2 made it happen sooner (which graphically means move to the left).

Hint: To remember which way to move the graph, set (x + k) = 0. The solution will tell you in which direction to move and by how much.
f (x  2): x  2 = 0 gives x = +2, move right 2 units.
f (x + 3): x + 3 = 0 gives x = 3, move left 3 units. 


Up to this point, we have only changed the "position" of the graph of the function.
Now, we will start changing "distorting" the shape of the graphs.
Dilations of Functions: kf (x) and f (kx) 
Vertical Stretch or Compression
k f (x) stretches/shrinks f (x) vertically 
"Multiply ycoordinates"
(x, y) becomes (x, ky)
"vertical dilation"

A vertical stretch pulls graph away from the xaxis.
A vertical compression (or shrink) squeezes graph toward the xaxis.
• if k > 1, the graph is vertically stretched by multiplying each of its ycoordinates by k.
• if 0 < k < 1 (a fraction), the graph vertically compressed (or shrunk) by multiplying each of its ycoordinates by k.
• if k should be negative, the vertical stretch or shrink is followed by a reflection across the xaxis.



Notice that the "roots" on the graph above stay in their same positions on the xaxis. The graph gets "taffy pulled" up and down from the locking root positions. The yvalues change. 
Horizontal Stretch or Compression
f (kx) stretches/shrinks f (x) horizontally 
"Divide xcoordinates"
(x, y) becomes (x/k, y)
"horizontal dilation"

A horizontal stretching pulls the graph away from yaxis
A horizontal compression (or shrink) squeezes the graph toward the yaxis.
• if k > 1, the graph is horizontally shrunk (or compressed) by dividing each of xcoordinate by k.
• if 0 < k < 1 (a fraction), the graph is horizontally stretched by dividing each of its xcoordinates by k.
• if k should be negative, the horizontal stretch or shrink is followed by a reflection in the yaxis. 


Notice that the "roots" on the graph above have now moved, but the yintercept stays in its same initial position for all graphs. The graph gets "taffy pulled" left and right from the locking yintercept. The xvalues change. 
Transformations of Function Graphs 

reflect f (x) over the xaxis

f (x) 
reflect f (x) over the yaxis 
f (x) + k 
shift f (x) up k units 
f (x)  k 
shift f (x) down k units 
f (x + k) 
shift f (x) left k units 
f (x  k) 
shift f (x) right k units 
k•f (x) 
multiply yvalues by k (k > 1 stretch, 0 < k < 1 shrink vertical) 
f (kx) 
divide xvalues by k (k > 1 shrink, 0 < k < 1 stretch horizontal) 
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