To determine the transformation that maps the function [tex]\( f(x) = (x - 7)^2 - 1 \)[/tex] to the function [tex]\( g(x) = (x + 6)^2 - 3 \)[/tex], we need to analyze the changes in the functions.
### Step-by-Step Analysis:
1. Examine the Inside of the Squared Terms:
- For [tex]\( f(x) \)[/tex], the term inside the square is [tex]\( (x - 7) \)[/tex].
- For [tex]\( g(x) \)[/tex], the term inside the square is [tex]\( (x + 6) \)[/tex].
To move from [tex]\( (x - 7) \)[/tex] to [tex]\( (x + 6) \)[/tex], we examine the changes:
- [tex]\( x - 7 \)[/tex] changes to [tex]\( x + 6 \)[/tex]. This represents a horizontal shift.
- By setting [tex]\( x - 7 = x + 6 \)[/tex], we solve for the shift:
[tex]\[
x - 7 = x + 6 \implies x + 13 = x \implies \text{shift left by 13 units}.
\][/tex]
2. Examine the Constant Terms:
- For [tex]\( f(x) \)[/tex], the constant term is [tex]\(-1\)[/tex].
- For [tex]\( g(x) \)[/tex], the constant term is [tex]\(-3\)[/tex].
To move from [tex]\(-1\)[/tex] to [tex]\(-3\)[/tex], we examine the changes:
- [tex]\(-1\)[/tex] changes to [tex]\(-3 \)[/tex], which represents a vertical shift.
- By determining the difference:
[tex]\[
-3 - (-1) = -3 + 1 = -2 \implies \text{shift down by 2 units}.
\][/tex]
### Conclusion:
Combining the horizontal and vertical shifts, we find that to map the function [tex]\( f(x) = (x - 7)^2 - 1 \)[/tex] onto [tex]\( g(x) = (x + 6)^2 - 3 \)[/tex], we need to apply the following transformation:
- Shift the graph left by 13 units.
- Shift the graph down by 2 units.
Therefore, the transformation that will map [tex]\( f(x) \)[/tex] onto [tex]\( g(x) \)[/tex] is left 13 units, down 2 units.
