The function $$ h $$ is defined as follows. $$ h(x)=3 x^{2}+1 $$ If the graph of $$ h $$ is translated vertically upward by 6 units, it becomes the graph of a function Find the expression for $$ f(x) $$. Note that the ALEKS graphing calculator may be helpful in checking your answer. $$ f(x)=\square $$

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To find the expression for the function $$ f(x) $$ after translating the graph of $$ h(x) $$ vertically upward by 6 units, we can follow these steps:

1. **Identify the original function**: The original function is given as:
   $$
   h(x) = 3x^2 + 1
   $$

2. **Understand vertical translation**: When a function is translated vertically upward by $$ k $$ units, the new function can be expressed as:
   $$
   f(x) = h(x) + k
   $$
   In this case, $$ k = 6 $$.

3. **Apply the translation**: We substitute $$ h(x) $$ into the equation for $$ f(x) $$:
   $$
   f(x) = h(x) + 6 = (3x^2 + 1) + 6
   $$

4. **Simplify the expression**: Now, we simplify the expression:
   $$
   f(x) = 3x^2 + 1 + 6 = 3x^2 + 7
   $$

Thus, the expression for $$ f(x) $$ after the vertical translation is:
$$
f(x) = 3x^2 + 7
$$
$$ f(x) = 3x^2 + 7 $$

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