The function $g(x)$ is a translation of $f(x)=(x+3)^2-10$. The axis of symmetry of $g(x)$ is 5 units to the right of $f(x)$. Which function could be $g(x)$?
A. $g(x)=(x-2)^2+k$
B. $g(x)=(x+8)^2+k$
C. $g(x)=(x-h)^2-5$
D. $g(x)=(x-h)^2-15$
Answer (2)
Find the axis of symmetry of f ( x ) = ( x + 3 ) 2 − 10 , which is x = − 3 .
Determine the axis of symmetry of g ( x ) , which is 5 units to the right of f ( x ) , so x = − 3 + 5 = 2 .
Check which of the given functions has an axis of symmetry of x = 2 .
g ( x ) = ( x − 2 ) 2 + k has an axis of symmetry of x = 2 , so the answer is g ( x ) = ( x − 2 ) 2 + k .
Explanation
Understanding the Problem We are given the function f ( x ) = ( x + 3 ) 2 − 10 and told that g ( x ) is a translation of f ( x ) . The axis of symmetry of g ( x ) is 5 units to the right of the axis of symmetry of f ( x ) . We need to determine which of the given functions could be g ( x ) .
Finding the Axis of Symmetry of f(x) The axis of symmetry of a quadratic function in the form f ( x ) = ( x − h ) 2 + k is given by x = h . For the function f ( x ) = ( x + 3 ) 2 − 10 , the axis of symmetry is x = − 3 .
Finding the Axis of Symmetry of g(x) Since the axis of symmetry of g ( x ) is 5 units to the right of the axis of symmetry of f ( x ) , the axis of symmetry of g ( x ) is x = − 3 + 5 = 2 .
Checking the Given Functions Now we need to check which of the given functions has an axis of symmetry of x = 2 .
g ( x ) = ( x − 2 ) 2 + k : The axis of symmetry is x = 2 . This is a possible solution.
g ( x ) = ( x + 8 ) 2 + k : The axis of symmetry is x = − 8 . This is not a possible solution.
g ( x ) = ( x − h ) 2 − 5 : The axis of symmetry is x = h . If h = 2 , then the axis of symmetry is x = 2 . This is a possible solution.
g ( x ) = ( x − h ) 2 − 15 : The axis of symmetry is x = h . If h = 2 , then the axis of symmetry is x = 2 . This is a possible solution.
Conclusion From the given options, g ( x ) = ( x − 2 ) 2 + k is a possible solution. Also, g ( x ) = ( x − h ) 2 − 5 and g ( x ) = ( x − h ) 2 − 15 are possible solutions if h = 2 .
Examples
Understanding translations of functions is crucial in various fields. For example, in physics, understanding how the graph of a projectile's height changes when the initial height is increased involves translating the original height function upwards. Similarly, in economics, shifting a cost function to the right can model the effect of increased production costs. These translations help in predicting and analyzing changes in real-world scenarios.
The function g ( x ) that could be a translation of f ( x ) with an axis of symmetry 5 units to the right is g ( x ) = ( x − 2 ) 2 + k . This has the correct symmetry at x = 2 . Other functions may also fit if adjusted correctly, but the simplest choice is clearly option A.
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