Welcome to Westonci.ca, the ultimate question and answer platform. Get expert answers to your questions quickly and accurately. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
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]
### 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]
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.