Let's analyze the function given: [tex]\( f(x) = |x + 8| - 3 \)[/tex].
1. Understanding the Parent Function:
The parent function here is [tex]\( f(x) = |x| \)[/tex], the absolute value function. This function creates a V-shaped graph with its vertex at the origin (0, 0).
2. Inside the Absolute Value:
- The expression inside the absolute value is [tex]\( x + 8 \)[/tex]. The general rule is that [tex]\( f(x + c) \)[/tex] represents a horizontal shift.
- In this case, [tex]\( x + 8 \)[/tex] means a horizontal shift to the left by 8 units.
Thus, we shift the graph of [tex]\( |x| \)[/tex] to the left by 8 units.
3. Outside the Absolute Value:
- The expression outside the absolute value is [tex]\(- 3\)[/tex]. The general rule here is that [tex]\( f(x) - k \)[/tex] represents a vertical shift.
- In this case, [tex]\(- 3\)[/tex] means a vertical shift down by 3 units.
Thus, after shifting the graph to the left by 8 units, we shift it down by 3 units.
Combining both transformations, the graph [tex]\( f(x) = |x + 8| - 3 \)[/tex] is:
- Shifted to the left by 8 units.
- Shifted down by 3 units.
Therefore, the correct transformation is:
The graph of [tex]\( f(x) = x \)[/tex] is shifted to the left 8 units, down 3 units.
The correct choice is:
3. The graph of [tex]\( f(x) = x \)[/tex] is shifted to the left 8 units, down 3 units.
