What value represents the vertical translation from the graph of the parent function [tex]f(x)=x^2[/tex] to the graph of the function [tex]g(x)=(x+5)^2+3[/tex] ?
A. -5
B. -3
C. 3
D. 5
Answer (2)
The vertical translation from the graph of f ( x ) = x 2 to the graph of g ( x ) = ( x + 5 ) 2 + 3 is the constant term added to the squared term in g ( x ) . In this case, the vertical translation is 3. Therefore, the answer is 3 .
Explanation
Understanding the Problem We are given the parent function f ( x ) = x 2 and the transformed function g ( x ) = ( x + 5 ) 2 + 3 . We want to find the vertical translation from the graph of f ( x ) to the graph of g ( x ) .
Identifying Vertical Translation The general form of a transformed quadratic function is a ( x − h ) 2 + k , where h represents the horizontal translation and k represents the vertical translation. In our case, g ( x ) = ( x + 5 ) 2 + 3 , so we can identify the vertical translation as the constant term added to the squared term.
Determining the Value In the function g ( x ) = ( x + 5 ) 2 + 3 , the vertical translation is represented by the value 3. This means the graph of f ( x ) = x 2 is translated vertically upwards by 3 units to obtain the graph of g ( x ) .
Examples
Understanding vertical translations is crucial in various fields. For example, in physics, when analyzing projectile motion, the vertical displacement of an object can be modeled using quadratic functions. If we know the initial height of the object, we can represent the height at any time t as h ( t ) = − 4.9 t 2 + v 0 t + h 0 , where h 0 is the initial height (vertical translation). Similarly, in economics, cost functions can be vertically translated to represent changes in fixed costs. If the original cost function is C ( x ) = x 2 , adding a constant, such as C ( x ) = x 2 + 1000 , represents an increase in fixed costs by $1000.
The vertical translation from the graph of the parent function f ( x ) = x 2 to the graph of the function g ( x ) = ( x + 5 ) 2 + 3 is 3. This indicates that the graph of f ( x ) is shifted upwards by 3 units to form g ( x ) . Therefore, the answer is 3 .
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