Transformations are 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 x-axis.
-f (x) reflects f (x) over the x-axis
Reflections are mirror images. Think of "folding" the graph over the x-axis.

 On a grid, you used the formula (x,y) → (x,-y) for a reflection in the x-axis, where the y-values were negated. Keeping in mind that y = f (x), we can write this formula as (x, f (x)) → (x, -f (x)).

Reflection over the y-axis.
f (-x) reflects f (x) over the y-axis
Reflections are mirror images. Think of "folding" the graph over the y-axis.

On a grid, you used the formula (x,y) → (-x,y) for a reflection in the y-axis, where the x-values 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
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 y-values of the function.
Translation horizontally (left or right)
f (x + k) translates f (x) left or right
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.
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.

Dilations of Functions:     kf (x)   and   f (kx)
Vertical Stretch or Compression (Shrink)

k f (x) stretches/shrinks f (x) vertically

"Multiply y-coordinates"
(x, y) becomes (x, ky)
"vertical dilation"

A vertical stretching is the stretching of the graph away from the x-axis
A vertical compression (or shrinking) is the squeezing of the graph toward the x-axis.
• if k > 1, the graph of y = k•f (x) is the graph of f (x) vertically stretched by multiplying each of its y-coordinates by k.
• if 0 < k < 1 (a fraction), the graph is f (x) vertically shrunk (or compressed) by multiplying each of its x-coordinates by k.
 • if k should be negative, the vertical stretch or shrink is followed by a reflection across the x-axis. Notice that the "roots" on the graph stay in their same positions on the x-axis. The graph gets "taffy pulled" up and down from the locking root positions. The y-values change.
Horizontal Stretch or Compression (Shrink)

f (kx) stretches/shrinks f (x) horizontally

"Divide x-coordinates"
(x, y) becomes (x/k, y)
"horizontal dilation"

A horizontal stretching is the stretching of the graph away from the y-axis
A horizontal compression (or shrinking) is the squeezing of the graph toward the y-axis.
• if k > 1, the graph of y = k•f (x) is the graph of f (x) horizontally shrunk (or compressed) by dividing each of its x-coordinates by k.
• if 0 < k < 1 (a fraction), the graph is f (x) horizontally stretched by dividing each of its x-coordinates by k.
• if k should be negative, the horizontal stretch or shrink is followed by a reflection in the y-axis.
 Notice that the "roots" on the graph have now moved, but the y-intercept stays in its same initial position for all graphs. The graph gets "taffy pulled" left and right from the locking y-intercept. The x-values change.

 Transformations of Function Graphs -f (x) reflect f (x) over the x-axis f (-x) reflect f (x) over the y-axis 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 y-values by k f (kx) divide x-values by k