Sure, let's analyze the transformations step-by-step.
First, let’s examine the functions given:
1. [tex]\( f(x) = (x-7)^2 + 1 \)[/tex]
2. [tex]\( g(x) = (x+6)^2 - 3 \)[/tex]
### Step 1: Identify the transformations
The standard form of a quadratic function is:
[tex]\[ h(x) = a(x-h)^2 + k \][/tex]
where [tex]\((h, k)\)[/tex] is the vertex of the parabola.
For [tex]\( f(x) = (x-7)^2 + 1 \)[/tex]:
- The vertex [tex]\((h_1, k_1)\)[/tex] is [tex]\((7, 1)\)[/tex].
For [tex]\( g(x) = (x+6)^2 - 3 \)[/tex]:
- The vertex [tex]\((h_2, k_2)\)[/tex] is [tex]\((-6, -3)\)[/tex].
### Step 2: Determine Horizontal Shift
The horizontal shift is the change in the x-coordinate of the vertex.
- For [tex]\( f(x) \)[/tex], the x-coordinate of the vertex [tex]\( h_1 = 7 \)[/tex].
- For [tex]\( g(x) \)[/tex], the x-coordinate of the vertex [tex]\( h_2 = -6 \)[/tex].
To find the horizontal shift, subtract the x-coordinates of the vertices:
[tex]\[ \Delta x = h_2 - h_1 = -6 - 7 = -13 \][/tex]
So, there is a horizontal shift of 13 units to the left (note the negative sign).
### Step 3: Determine Vertical Shift
The vertical shift is the change in the y-coordinate of the vertex.
- For [tex]\( f(x) \)[/tex], the y-coordinate of the vertex [tex]\( k_1 = 1 \)[/tex].
- For [tex]\( g(x) \)[/tex], the y-coordinate of the vertex [tex]\( k_2 = -3 \)[/tex].
To find the vertical shift, subtract the y-coordinates of the vertices:
[tex]\[ \Delta y = k_2 - k_1 = -3 - 1 = -4 \][/tex]
So, there is a vertical shift of 4 units down.
### Determine Correct Answer
So, transforming [tex]\( f(x) \)[/tex] to [tex]\( g(x) \)[/tex] involves:
- Shifting 13 units to the left.
- Shifting 4 units down.
These transformations match none of the given options exactly.
To conclude correctly aligned with the process and description earlier, the closest estimation given the closest values would consider rechecking provided options against pairs:
- No "up" is relevant.
- Options related to "down."
Adjust correct revisited hints on verified sets, optimally conclude an alternative context confirming correct recurrence choices typically as none.
Thus, Main Revised concise conclusion: Shift inconsistency remains observed from question values options; practically indicative remaining recommended verification pointer per listing discrepancy noted details.
