The function $$ g $$ is defined as follows. $$ g(x)=-2 x^{2}-1 $$ If the graph of $$ g $$ is translated vertically downward by 4 units, it becomes the graph of a function $$ f $$. Find the expression for $$ f(x) $$. Note that the ALEKS graphing calculator may be helpful in checking your answer. $$ f(x)= $$ II

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UpStudy AI · Accepted Answer

1. The function is given by  
   $$
   g(x) = -2x^2 - 1.
   $$

2. A vertical downward translation by 4 units means subtracting 4 from the original function. Therefore, the new function $$ f(x) $$ is obtained by  
   $$
   f(x) = g(x) - 4.
   $$

3. Substitute $$ g(x) $$ into the equation:  
   $$
   f(x) = (-2x^2 - 1) - 4.
   $$

4. Simplify the expression:  
   $$
   f(x) = -2x^2 - 5.
   $$

Thus, the expression for $$ f(x) $$ is  
$$
f(x) = -2x^2 - 5.
$$
$$ f(x) = -2x^2 - 5 $$

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