| The transformations you have seen in the past can besides be used to motility and resize graphs of functions. Nosotros will be examining the post-obit changes to f (x):
- f (x), f (-x),f (ten) + chiliad, f (x + thousand), kf (x), f (kx)
reflections translations dilations | Reflections of Functions: -f (ten) and f (-ten) | |  Reflection over the x-centrality.
-f (x) reflects f (10) over the ten-axis
|  | Vertical Reflection:
Reflections are mirror images. Think of "folding" the graph over the 10-axis. | On a grid, you used the formula
(x,y) → (ten,-y) for a reflection in the
x-axis, where the y-values were negated. Keeping in listen that y = f (10),
nosotros tin can write this formula as
(x, f (ten)) → (x, -f (x)). |  |
| | Reflection over the y-axis.
f (-x) reflects f (x) over the y-axis |  | | Horizontal Reflection:
Reflections are mirror images. Recollect of "folding" the graph over the y-axis. On a grid, you used the formula (x,y) → (-ten,y) for a reflection in the y-axis, where the x-values were negated. Keeping in mind that
y = f (x), nosotros tin can write this formula as
(10, f (10)) → (-ten, f (-x)). 
| | Translations of Functions: f (x) + k and f (ten + k) | | Translation vertically (upward or downwardly)
f (x) + yard translates f (x) up or down |  Changes occur "outside" the function
(affecting the y-values). | Vertical Shift:
This translation is a "slide" straight upward or down.
• if k > 0, the graph translates upward k units.
• if grand < 0, the graph translates downward thou units. On a filigree, you lot used the formula (ten,y) → (x,y + grand) to move a figure upward or down. Keeping in | mind that y = f (x), we can write this formula every bit (ten, f (10)) → (x, f (ten) + k ).
Think, y'all are adding the value
of k to the y-values of the function. |  | | | Translation horizontally (left or correct)
f (10 + grand) translates f (x) left or right | 
Changes occur "within" the role
(affecting the x-axis). | | Horizontal Shift:
This translation is a "slide" left or correct.
• if k > 0, the graph translates to the left k units.
• if k < 0, the graph translates to the right grand units. 
This one will not be obvious from the patterns you lot previously used when translating points.
thousand positive moves graph left
one thousand negative moves graph right
A horizontal shift means that every point (ten,y) on the graph of f (10) is transformed to (x - k, y) or (x + 1000, y) on the graphs of y = f (10 + k) or y = f (ten - k) respectively.
Await carefully as this can be very confusing!
| | Hint: To recollect which way to move the graph, set (x + thousand) = 0. The solution will tell you in which management to move and by how much.
f (x - two): ten - 2 = 0 gives x = +2, move correct two units.
f (10 + 3): x + 3 = 0 gives ten = -three, move left three units. |  | | Upwards to this point, nosotros have only changed the "position" of the graph of the function.
At present, we volition start irresolute "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 (ten) 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 m > i , 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 < g < i (a fraction), the graph is f (ten) vertically shrunk (or compressed) by multiplying each of its y-coordinates by m.
| • if k should be negative, the vertical stretch or shrink is followed by a reflection beyond the x-centrality.
Observe that the "roots" on the graph stay in their same positions on the x-axis. The graph gets "taffy pulled" upward and down from the locking root positions. The y-values change. |  | | | Horizontal Stretch or Compression (Shrink)
f (kx) stretches/shrinks f (x) horizontally |  "Dissever x-coordinates"
(x, y) becomes (x/one thousand, 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 1000 > 1 , the graph of y = f (m•10) is the graph of f (x) horizontally shrunk (or compressed) by dividing each of its x-coordinates by g.
• if 0 < m < 1 (a fraction), the graph is f (x) horizontally stretched past dividing each of its ten-coordinates past k.
• if one thousand should be negative, the horizontal stretch or compress is followed past 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 | | | reflect f (x) over the x-axis | | f (-x) | reflect f (x) over the y-axis | | f (x) + k | shift f (x) upwardly k units | | f (x) - thousand | shift f (x) downwardly k units | | f (ten + g) | shift f (x) left k units | | f (x - k) | shift f (x) correct k units | | k•f (x) | multiply y-values by k (yard > 1 stretch, 0 < thousand < one shrink vertical) | | f (kx) | divide ten-values by m(1000 > one compress, 0 < k < ane stretch horizontal) |  Note: The re-posting of materials (in part or whole) from this site to the Net is copyright violation
and is not considered "fair use" for educators. Please read the "Terms of Utilise". |
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