Describe in words the transformations to the function:

[tex]f(x)=3|x+1|-3[/tex]

Horizontal Shift: $\square$
-example "left 2" or "right 1"

Vertical Shift: $\square$
-example "down 2" or "up 3"

Horizontal Stretch/Compression: $\square$ -example "stretch 5"

Vertical Stretch/Compression: $\square$ -example "compress 3"

Horizontal Reflection: ( [tex]y[/tex]-axis Reflection) $\square$ -example "yes" or a reflection " 0 " for no reflection

Vertical Reflection ( [tex]x[/tex]-axis Reflection): $\square$ -example "yes" or a reflection "0" for no reflection. If the category does not apply, type in 0.

The function f ( x ) = 3∣ x + 1∣ − 3 undergoes several transformations: a horizontal shift left by 1 unit, a vertical shift down by 3 units, and a vertical stretch by a factor of 3. There are no horizontal or vertical reflections. So, the final transformations are left 1, down 3, stretch 3, no, and 0.
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Answered by Anonymous | 2025-07-07

The function f ( x ) = 3∣ x + 1∣ − 3 undergoes a horizontal shift to the left by 1 unit.
It also experiences a vertical shift downwards by 3 units.
The function is vertically stretched by a factor of 3.
There are no horizontal or vertical reflections.
Therefore, the transformations are left 1, down 3, stretch 3, no, and 0. ​

Explanation

Analyzing the Function The function we are analyzing is f ( x ) = 3∣ x + 1∣ − 3 . We need to identify and describe the transformations applied to the parent absolute value function, y = ∣ x ∣ . These transformations include horizontal and vertical shifts, stretches/compressions, and reflections.

Identifying Horizontal Shift The term inside the absolute value is x + 1 . This indicates a horizontal shift. Since it's x + 1 , the shift is to the left by 1 unit.

Identifying Vertical Shift The constant term outside the absolute value is − 3 . This indicates a vertical shift. Since it's − 3 , the shift is downward by 3 units.

Identifying Vertical Stretch/Compression The coefficient of the absolute value term is 3 . This indicates a vertical stretch. Since the coefficient is 3 , the function is stretched vertically by a factor of 3.

Checking for Horizontal Reflection There is no negative sign inside the absolute value directly affecting x . Therefore, there is no horizontal reflection (reflection about the y -axis).

Checking for Vertical Reflection There is no negative sign in front of the absolute value term. Therefore, there is no vertical reflection (reflection about the x -axis).

Stating the Transformations In summary, the transformations are:



Horizontal Shift: Left 1
Vertical Shift: Down 3
Vertical Stretch: Stretch 3
Horizontal Reflection: No
Vertical Reflection: No

Examples
Understanding transformations of functions is crucial in many fields. For example, in physics, understanding how graphs of motion change with different initial conditions or forces involves transformations. In economics, shifts in supply and demand curves can be described using similar transformations. In computer graphics, transformations are used to manipulate objects in 2D and 3D space. For instance, if you are designing a game and want to move an object 5 units to the left and 3 units down, and also make it three times bigger, you would apply these transformations to the object's coordinates.

Answered by GinnyAnswer | 2025-07-07