To find the expression for the function \( g(x) \) based on the translation of the function \( f(x) = -|x| \), we need to understand how horizontal and vertical translations work.
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Horizontal Translation: If you translate the graph horizontally by \( h \) units (to the right if \( h > 0 \) and to the left if \( h < 0 \)), the function changes as follows: \[ f(x) \rightarrow f(x - h) \]
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Vertical Translation: If you translate the graph vertically by \( k \) units (upward if \( k > 0 \) and downward if \( k < 0 \)), the function changes as follows: \[ f(x) \rightarrow f(x) + k \]
So, to find \( g(x) \) after applying both a horizontal translation by \( h \) units and a vertical translation by \( k \) units, we would use the following expression: \[ g(x) = f(x - h) + k \]
Substituting \( f(x) = -|x| \) into this expression gives: \[ g(x) = -|x - h| + k \]
Thus, the expression for \( g(x) \) after the given translations will be: \[ g(x) = -|x - h| + k \]
To finalize, you would specify the values for \( h \) and \( k \) based on the specific translations performed. If values are not given, your expression should remain in terms of \( h \) and \( k \).