To shift the graph of \( f(x) = 2x^2 \) left 3 units, you replace \( x \) with \( x + 3 \). To shift the graph up 2 units, you add 2 to the entire function.
Starting with \( f(x) \):
- Shift left 3 units: \( f(x + 3) = 2(x + 3)^2 \)
- Shift up 2 units: \( g(x) = 2(x + 3)^2 + 2 \)
Now, we can simplify \( g(x) \):
\[ g(x) = 2(x + 3)^2 + 2 = 2(x^2 + 6x + 9) + 2 = 2x^2 + 12x + 18 + 2 = 2x^2 + 12x + 20 \]
Therefore, the equation of the function \( g(x) \) is:
\[ g(x) = 2(x + 3)^2 + 2 \]
In its expanded form, it can also be written as:
\[ g(x) = 2x^2 + 12x + 20 \]