To determine which transformation results in the doubling of the cost function [tex]\( f(x) = 2x^2 + 6000 \)[/tex], we need to examine how the graph of this function changes when the cost doubles.
### Step-by-Step Solution:
1. Original Cost Function:
The given cost function is:
[tex]\[
f(x) = 2x^2 + 6000
\][/tex]
This represents the cost of manufacturing [tex]\( x \)[/tex] refrigerators.
2. Doubling the Cost Function:
If the cost doubles, the entire cost function needs to be multiplied by 2. Therefore, the new cost function becomes:
[tex]\[
\text{new } f(x) = 2 \times (2x^2 + 6000)
\][/tex]
3. Simplification of the New Function:
Simplifying the new function gives:
[tex]\[
\text{new } f(x) = 4x^2 + 12000
\][/tex]
4. Understanding the Transformation:
To find the appropriate transformation, compare the new function [tex]\( 4x^2 + 12000 \)[/tex] to the original function [tex]\( 2x^2 + 6000 \)[/tex].
- The term [tex]\( 4x^2 \)[/tex] indicates that each [tex]\( x^2 \)[/tex] term has been multiplied by 4 (originally it was [tex]\( 2x^2 \)[/tex], and now it is [tex]\( 4x^2 \)[/tex]).
- The constant term has doubled from 6000 to 12000.
Therefore, the overall effect is that the entire cost function has stretched vertically by a factor of 2.
5. Resulting Transformation:
When you multiply the entire function by a constant factor (greater than 1), the graph of the function is vertically stretched by that factor.
In this case:
[tex]\[
2 \times (2x^2 + 6000) = 4x^2 + 12000
\][/tex]
represents a vertical stretch by a factor of 2.
Thus, the transformation that results from the doubling of the cost function is a vertical stretch.
### Final Answer:
[tex]\[
\boxed{B. \text{vertical stretch}}
\][/tex]
