Sure, let's go through the steps to graph the function [tex]\( g(x) = |x + 2| - 4 \)[/tex] using the parent function [tex]\( f(x) = |x| \)[/tex].
### Step-by-Step Solution
#### Step 1: Determine the transformations
The function [tex]\( g(x) = |x + 2| - 4 \)[/tex] involves two transformations applied to the parent function [tex]\( f(x) = |x| \)[/tex]:
1. Horizontal Shift: [tex]\( |x + 2| \)[/tex] means the graph shifts left by 2 units.
2. Vertical Shift: The [tex]\(- 4\)[/tex] at the end of the function indicates a downward shift by 4 units.
#### Step 2: Identify the piecewise nature of the function
The absolute value function [tex]\( f(x) = |x| \)[/tex] is piecewise, and so is [tex]\( g(x) \)[/tex]:
- For [tex]\( x \geq -2 \)[/tex], [tex]\( |x + 2| = x + 2 \)[/tex].
- For [tex]\( x < -2 \)[/tex], [tex]\( |x + 2| = -(x + 2) \)[/tex].
By substituting [tex]\( |x + 2| \)[/tex] in the equation [tex]\( g(x) = |x + 2| - 4 \)[/tex], we derive two cases:
- When [tex]\( x \geq -2 \)[/tex]:
[tex]\[ g(x) = (x + 2) - 4 = x - 2 \][/tex]
- When [tex]\( x < -2 \)[/tex]:
[tex]\[ g(x) = -(x + 2) - 4 = -x - 6 \][/tex]
#### Step 3: Determine key points for each ray
We need at least two points for each piece of the piecewise function to graph the rays.
For the ray when [tex]\( x \geq -2 \)[/tex]:
[tex]\[
g(x) = x - 2
\][/tex]
- Point when [tex]\( x = -2 \)[/tex]:
[tex]\[
g(-2) = -2 - 4 = -4 \quad \text{(Point: } (-2, -4) \text{)}
\][/tex]
- Point when [tex]\( x = 0 \)[/tex]:
[tex]\[
g(0) = 0 - 2 = -2 \quad \text{(Point: } (0, -2) \text{)}
\][/tex]
For the ray when [tex]\( x < -2 \)[/tex]:
[tex]\[
g(x) = -x - 6
\][/tex]
- Point when [tex]\( x = -4 \)[/tex]:
[tex]\[
g(-4) = -(-4) - 6 = 4 - 6 = -2 \quad \text{(Point: } (-4, -2) \text{)}
\][/tex]
- Point when [tex]\( x = -2 \)[/tex]:
[tex]\[
g(-2) = -(-2) - 6 = 2 - 6 = -4 \quad \text{(Point: } (-2, -4) \text{)}
\][/tex]
Notice that the point [tex]\( (-2, -4) \)[/tex] is common in both rays, serving as the vertex where the rays meet.
#### Step 4: Write the points for each ray
Based on the calculations, the points for each ray are:
- Ray for [tex]\( x \geq -2 \)[/tex]:
[tex]\[
(-2, -4), (0, -2)
\][/tex]
- Ray for [tex]\( x < -2 \)[/tex]:
[tex]\[
(-4, -2), (-2, -4)
\][/tex]
You can now use these points to graph the transformed function [tex]\( g(x) = |x + 2| - 4 \)[/tex].
