12 terms

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).

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).

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).

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).

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.

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)

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)

(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)

(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).

(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).

(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

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.

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.