Question 20 (5 points)
Which function is the result of translating $y=x^2$ downward by 3 units and to the left by 4 units?
A) $y=(x-4)^2+3$
B) $y=(x+3)^2-4$
C) $y=(x-3)^2+4$
D) $y=(x+4)^2-3$
Answer (1)
Translating y = x 2 downward by 3 units results in y = x 2 − 3 .
Translating y = x 2 − 3 to the left by 4 units results in y = ( x + 4 ) 2 − 3 .
The final function is y = ( x + 4 ) 2 − 3 .
The answer is y = ( x + 4 ) 2 − 3 .
Explanation
Understanding the Problem We are given the function y = x 2 and asked to translate it downward by 3 units and to the left by 4 units. We need to find the resulting function.
Downward Translation To translate a function y = f ( x ) downward by k units, we replace y with y + k , which gives us y + k = f ( x ) , or y = f ( x ) − k . In our case, f ( x ) = x 2 and k = 3 , so translating y = x 2 downward by 3 units gives us y = x 2 − 3 .
Leftward Translation To translate a function y = f ( x ) to the left by h units, we replace x with x + h , which gives us y = f ( x + h ) . In our case, h = 4 , so translating y = x 2 − 3 to the left by 4 units gives us y = ( x + 4 ) 2 − 3 .
Final Answer Therefore, the function resulting from translating y = x 2 downward by 3 units and to the left by 4 units is y = ( x + 4 ) 2 − 3 . This corresponds to option D.
Examples
Understanding transformations of functions is crucial in many fields. For example, in physics, understanding how graphs of motion change when time or distance is shifted helps in analyzing real-world scenarios. Similarly, in economics, shifting cost or revenue curves helps in understanding the impact of taxes or subsidies. In computer graphics, transformations are used to manipulate objects on the screen. The translation of functions is a fundamental concept that builds the foundation for more complex transformations and applications.
