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

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