1. The problem states that the function $h$ is given by: $$h(x) = 4x^{2} - 9$$ 2. We are asked to find the function $f$ obtained by translating the graph of $h$ vertically upward by 3 units. 3. A vertical translation upward by 3 units means we add 3 to the entire function: $$f(x) = h(x) + 3$$ 4. Substitute $h(x)$ into the expression: $$f(x) = (4x^{2} - 9) + 3$$ 5. Simplify the expression: $$f(x) = 4x^{2} - 9 + 3 = 4x^{2} - 6$$ 6. Therefore, the expression for $f(x)$ after the vertical translation is: $$f(x) = 4x^{2} - 6$$