|  | Transformations of Functions
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| To review basic transformations, see Symmetry, Reflections, Translations, Dilations and Rotations. We have seen the transformations used in past courses can be used to
move and resize graphs of functions. We examined the following changes to f (x):
- f (x), f (-x), f (x) + k, f (x + k), kf (x), f (kx)
reflections translations dilations This page is a summary of all of the function transformation we have investigated.
For more information on each transformation, follow the links within each section below. Reflections of Functions: -f (x) and f (-x) | Reflection over the x-axis.
-f (x) reflects f (x) over the x-axis
|
| Vertical Reflection:
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 |
| Horizontal Reflection:
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 | Changes occur "outside" the function
(affecting the y-values). | 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 y-values of the function. | | | Translation horizontally (left or right)
f (x + k) translates f (x) left or right |
Changes occur "inside" the function
(affecting the x-axis).
| 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. |
| | Transformation that "distort" (change) the "shape" of the function. | 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 y-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 = f (k•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 | | Notation | Changes to f (x) | Coordinate Change | | reflection over the x-axis | (x, y) → (x, -y) | f (-x) | reflection over the y-axis | (x, y) → (-x, y) | f (x) + k | vertical shift up k units | (x, y) → (x, y + k) | f (x) - k | vertical shift down k units | (x, y) → (x, y - k) | f (x + k) | horizontal shift left k units | (x, y) → (x - k, y) | f (x - k) | horizontal shift right k units | (x, y) → (x + k, y) | k•f (x) | vertical stretch: | k | > 1
vertical compression: 0 < | k | < 1 | (x, y) → (x, k•y) | f (kx) | horizontal compression: | k | > 1
horizontal stretch: 0 < | k | < 1 | (x, y) → (x/k, y) |
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Changes occur "outside" the function
