Let's analyze the transformation of the parent function [tex]\( f(x) = \sqrt[3]{x} \)[/tex].
The transformation given is [tex]\( g(x) = f(x+2) - 4 \)[/tex].
This involves two steps:
1. Horizontal Shift: The term [tex]\( f(x+2) \)[/tex] translates the graph of [tex]\( f(x) = \sqrt[3]{x} \)[/tex] horizontally to the left by 2 units.
2. Vertical Shift: The subtraction of 4, [tex]\( -4\)[/tex], translates the graph vertically down by 4 units.
To understand this transformation, let's break it down:
### Step 1: Horizontal Shift
- Original function: [tex]\( f(x) = \sqrt[3]{x} \)[/tex]
- Transform function horizontally by shifting 2 units to the left:
[tex]\[ f(x + 2) = \sqrt[3]{x + 2} \][/tex]
### Step 2: Vertical Shift
- Now, take the horizontally shifted function and shift it vertically down by 4 units:
[tex]\[ g(x) = f(x + 2) - 4 = \sqrt[3]{x + 2} - 4 \][/tex]
Thus, the transformed function [tex]\( g(x) \)[/tex] is:
[tex]\[ g(x) = \sqrt[3]{x + 2} - 4 \][/tex]
### Graphical Description:
1. Horizontal Shift to the Left:
- The graph of [tex]\( f(x) = \sqrt[3]{x} \)[/tex] normally passes through points like [tex]\((0, 0)\)[/tex], [tex]\((1, 1)\)[/tex], and [tex]\((-1, -1)\)[/tex].
- Shifting it to the left by 2 units means the new graph will pass through [tex]\((-2, 0)\)[/tex], [tex]\((-1, 1)\)[/tex], and [tex]\((-3, -1)\)[/tex].
2. Vertical Shift Downward:
- After the horizontal shift, adjust the graph downward by 4 units.
- So the graph points now are shifted as: [tex]\((-2, -4)\)[/tex], [tex]\((-1, -3)\)[/tex], and [tex]\((-3, -5)\)[/tex].
### Understanding Graph Behavior:
- The basic shape of [tex]\( g(x) \)[/tex] remains the same as the cube root function [tex]\( f(x) \)[/tex].
- Cube root functions are characterized by an S-shaped curve where [tex]\( f(x) = \sqrt[3]{x} \)[/tex] rises more slowly and extends infinitely in both directions.
- Each point on the curve of [tex]\( f(x) \)[/tex] will be shifted left by 2 units and then down by 4 units for [tex]\( g(x) \)[/tex].
Thus, the graph of [tex]\( g(x) = \sqrt[3]{x + 2} - 4 \)[/tex] is the same basic S-shaped graph of [tex]\( \sqrt[3]{x} \)[/tex], but shifted 2 units to the left and 4 units down.
For visualization purposes, consider key transformed points and plot these changes accordingly to illustrate the described transformations.
