
The transformations you have seen in the past can also be used to move and resize 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) 
Reflection over the xaxis.
f (x) reflects f (x) over the xaxis


Vertical Reflection:
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. Keeping in mind that y = f (x),
we can write this formula as
(x, f (x)) → (x, f (x)). 


Reflection over the yaxis.
f (x) reflects f (x) over the yaxis 

Horizontal Reflection:
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. Keeping in mind that
y = f (x), we can write this formula as
(x, f (x)) → (x, f (x)).

Translations of Functions: f (x) + k and f (x + k) 
Translation vertically (upward or downward)
f (x) + k translates f (x) up or down 
Changes occur "outside" the function
(affecting the yvalues). 
Vertical Shift:
This translation is a "slide" straight up or down.
• if k > 0, the graph translates upward k units.
• if k < 0, the graph translates downward k units.
On a grid, you used the formula (x,y) → (x,y + k) to move a figure upward or downward. Keeping in
mind that y = f (x), we can write this formula as (x, f (x)) → (x, f (x) + k).
Remember, you are adding the value
of k to the yvalues of the function.



Translation horizontally (left or right)
f (x + k) translates f (x) left or right 
Changes occur "inside" the function
(affecting the xaxis). 
Horizontal Shift:
This translation is a "slide" left or right.
• if k > 0, the graph translates to the left k units.
• if k < 0, the graph translates to the right k units.
This one will not be obvious from the patterns you previously used when translating points.
k positive moves graph left
k negative moves graph right
A horizontal shift means that every point ( x,y) on the graph of f ( x) is transformed to ( x  k, y) or ( x + k, y) on the graphs of y = f ( x + k) or y = f ( x  k) respectively.
Look carefully as this can be very confusing!

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 (Shrink)
k f (x) stretches/shrinks f (x) vertically 
"Multiply ycoordinates"
(x, y) becomes (x, ky)
"vertical dilation"

A vertical stretching is the stretching of the graph away from the xaxis
A vertical compression (or shrinking) is the squeezing of the graph toward the xaxis.
• if k > 1, the graph of y = k•f (x) is the graph of f (x) vertically stretched by multiplying each of its ycoordinates by k.
• if 0 < k < 1 (a fraction), the graph is f (x) vertically shrunk (or compressed) 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 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 (Shrink)
f (kx) stretches/shrinks f (x) horizontally 
"Divide xcoordinates"
(x, y) becomes (x/k, y)
"horizontal dilation"

A horizontal stretching is the stretching of the graph away from the yaxis
A horizontal compression (or shrinking) is the squeezing of the graph toward the yaxis.
• if k > 1, the graph of y = f (k•x) is the graph of f (x) horizontally shrunk (or compressed) by dividing each of its xcoordinates by k.
• if 0 < k < 1 (a fraction), the graph is f (x) 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 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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