Without graphing, identify the verx, axis of symmetry, and transformations from the parent function f(x)= absolute value of x

y= 4- absolute value of x-2

the four moves it vertically 4 units.

the x-2 shifts it to the left, two units

Thanks I hope it's right

y=3|x+6|

To identify the vertex, axis of symmetry, and transformations from the parent function f(x) = |x| to the given function y = 4 - |x - 2|, we can follow these steps:

Step 1: Identify the basic parent function: f(x) = |x|

The basic parent function is f(x) = |x|. It is a V-shaped graph centered at the origin.

Step 2: Identify the transformations:

The given function is y = 4 - |x - 2|, which has the following transformations:

1. Vertical shift: The "+4" term in y = 4 - |x - 2| indicates a vertical shift upward by 4 units from the parent function. The graph will be shifted 4 units above the x-axis.

2. Horizontal shift: The "-2" inside the absolute value function |x - 2| indicates a horizontal shift to the right by 2 units from the parent function. The graph will be shifted 2 units to the right.

It is important to note that the horizontal shift is opposite to the sign inside the absolute value function. If it were "+2" instead of "-2", it would indicate a horizontal shift to the left.

Step 3: Identify the vertex and the axis of symmetry:

Since the basic parent function is symmetric about the y-axis, any horizontal shift does not affect the axis of symmetry. The axis of symmetry remains the vertical line x = 0.

To find the vertex, we need to determine the point at which the minimum or maximum value occurs. In the case of absolute value functions, the vertex occurs when the expression inside the absolute value function is zero.

In the given function y = 4 - |x - 2|, the expression x - 2 = 0 when x = 2. Therefore, the vertex occurs at the point (2, 4).

Summary:
- The vertex of the given function is (2, 4).
- The axis of symmetry remains the vertical line x = 0.
- The transformations from the parent function f(x) = |x| are a vertical shift upward by 4 units and a horizontal shift to the right by 2 units.