The function \( f \) is defined as follows. \[ f(x)=3 x^{2}+2 \] If the graph of \( f \) is translated vertically downward by 5 units, it becomes the graph of a function \( g \). Find the expression for \( g(x) \). Note that the ALEKS graphing calculator may be helpful in checking your answer. \( g(x)= \)

To find the expression for the function \( g(x) \) after translating the graph of \( f(x) \) vertically downward by 5 units, we can follow these steps: 1. **Identify the original function**: The original function is given as: \[ f(x) = 3x^2 + 2 \] 2. **Understand the translation**: Translating a graph vertically downward by \( k \) units means we subtract \( k \) from the function. In this case, \( k = 5 \). 3. **Apply the translation**: To find \( g(x) \), we subtract 5 from \( f(x) \): \[ g(x) = f(x) - 5 \] 4. **Substitute \( f(x) \) into the equation**: \[ g(x) = (3x^2 + 2) - 5 \] 5. **Simplify the expression**: \[ g(x) = 3x^2 + 2 - 5 = 3x^2 - 3 \] Thus, the expression for \( g(x) \) is: \[ g(x) = 3x^2 - 3 \]