Let's analyze the function [tex]\( f(x) = 2|x| - 3 \)[/tex] by comparing it to its parent function [tex]\( f(x) = |x| \)[/tex].
1. Parent Function: The parent function is [tex]\( f(x) = |x| \)[/tex]. This is a basic absolute value function that forms a "V" shape with its vertex at the origin (0, 0).
2. Vertical Stretch: The term [tex]\( 2|x| \)[/tex] indicates a vertical stretch by a factor of 2. In other words, each y-value of [tex]\( |x| \)[/tex] is multiplied by 2. Therefore, the graph of [tex]\( 2|x| \)[/tex] is stretched vertically, making it steeper than [tex]\( |x| \)[/tex].
3. Vertical Translation: The term [tex]\(-3\)[/tex] represents a vertical translation. Specifically, it shifts the entire graph downward by 3 units. Essentially, you move each point on the graph of [tex]\( 2|x| \)[/tex] down by 3 units.
Now we consider whether there are any horizontal shifts or other transformations:
- There are no horizontal shifts: nothing in the equation suggests moving the graph left or right.
- There are no reflections or any other unusual transformations.
In conclusion:
- The function [tex]\( f(x) = 2|x| - 3 \)[/tex] undergoes a vertical stretch and a vertical translation downward by 3 units.
Therefore, the correct answer to the question of what type of transformation is represented is:
Vertical translation
