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Precalculus-Shifting, Reflecting, and Stretching Graphs
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How do you describe an upward vertical translation (shift) "c" in function notation?
h(x)=f(x) + c
Where f(x) is one of the original parent functions before any transformation. The entire graph moves up by "c" amount of units. Let "c" be a positive real number.
(x,y,) -> (x, y+c) add c to y-coordinate
Example:
In this diagram, the parent function is f(x)=x² (blue).
Then it is translated (shifted) vertically up by two units, which is represented by the function g(x)=x²+2 (red).
How do you describe a downward vertical translation (shift) "c" in function notation?
h(x)=f(x) - c
Where f(x) is one of the original parent functions before any transformation. The entire graph moves down by "c" amount of units. Let "c" be a positive real number.
(x,y,) -> (x, y-c) subtract c from y-coordinate
Example:
In this diagram, the parent function is f(x)=x² (blue).
Then it is translated (shifted) vertically down by three units, which is represented by the function t(x)=x²-3 (green).
How do you describe a right horizontal translation (shift) "c" in function notation?
h(x)=f(x-c)
Where f(x) is one of the original parent functions before any transformation. The entire graph moves to the right by "c" amount of units. Let "c" be a positive real number.
(x,y) -> (x+c, y) add c to the x-coordinate
Example:
In this diagram, the parent function is f(x)=x² (blue).
Then it is horizontally translated (shifted) to the right by three units, which is represented by the function t(x)=(x-3)² (green).
How do you describe a left horizontal translation (shift) "c" in function notation?
h(x)=f(x+c)
Where f(x) is one of the original parent functions before any transformation. The entire graph moves to the left by "c" amount of units. Let "c" be a positive real number.
(x,y) -> (x-c, y) subtract c from the x-coordinate
Example:
In this diagram, the parent function is f(x)=x² (blue).
Then it is horizontally translated (shifted) to the left by two units, which is represented by the function g(x)=(x+2)² (red).
How do you describe a reflection in the x-axis in function notation?
h(x)=-f(x)
Where f(x) is one of the original parent functions before any transformation. The entire graph reflects across the x-axis, creating a mirror image of the parent function.
(x,y) -> (x,-y) multiply the y-coordinate by -1
Example:
In this diagram, the parent function is f(x)=x² (blue).
Then it is reflected (mirrored) in the x-axis. Both functions:
g(x)=-2x² (red)
and
t(x)=(-1/3)x² (green)
are reflections because of the -2 and (-1/3), respectively.
How do you describe a reflection in the y-axis in function notation?
h(x)=f(-x)
Where f(x) is one of the original parent functions before any transformation. The entire graph reflects across the y-axis, creating a mirror image of the parent function.
(x,y) -> (-x,y) multiply the x-coordinate by -1
Example:
In this diagram, the parent function is f(x)=√x (green dotted).
Then it is reflected (mirrored) in the y-axis. The resulting function is g(x)=√-x (blue)
How do you describe a vertical stretch in function notation?
h(x)=cf(x), where c>1, and "c" is multiplied with each of the y-coordinates.
(x,y) -> (x, cy) multiply y-coordinate by c
Example:
In this diagram, the parent function is f(x)=x² (blue)
It is then multiplied by "c," which is 2.
2 is greater than 1, so it is a vertical stretch. The resulting function is g(x)=2x² (red)
How do you describe a vertical shrink in function notation?
h(x)=cf(x), where 0<c<1, and "c" is multiplied with each of the y-coordinates.
(x,y) -> (x, cy) multiply y-coordinate by c
Example:
In this diagram, the parent function is f(x)=x² (blue)
It is then multiplied by "c," which is 1/3.
1/3 is between 0 and 1, so it is a vertical shrink. The resulting function is t(x)=(1/3)x² (green)
How do you describe a horizontal stretch in function notation?
h(x)=f(cx), where 0<c<1, and each x-coordinate is divided by "c."
(x,y) -> (x/c, y) divide x-coordinate by c
Example:
In this diagram, the parent function is f(x) (blue) and the resulting function is f(cx) (red).
How do you describe a horizontal shrink in function notation?
h(x)=f(cx), where c>1, and each x-coordinate is divided by "c."
(x,y) -> (x/c, y) divide x-coordinate by c
Example:
In this diagram, the parent function is f(x) (blue) and the resulting function is f(cx) (red).
What is the general function to show all types of transformations?
g(x) = Af(Bx + H) + K
A: vertical stretch or shrink
-A: reflection in the x-axis
B: horizontal stretch or shrink
-B: reflection in the y-axis
H: horizontal shift left or right
K: vertical shift up or down
When given a function, in what order must you complete each transformation?
g(x) = Af(Bx + H) + K
1. Shift: horizontal left or right (H)
2. Stretch/Shrink: horizontal (B)
3. Stretch/Shrink: vertical (A)
4. Reflection: about y-axis. (-B)
5. Reflection: about the x-axis. (-A)
6. Shift: vertical up or down. (K)
Steps 2-5 can be done in a different order, but you must always do horizontal shifts first and vertical shifts last.
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