Answer: g(x)= 2x² - 12x + 21.
Step-by-step explanation: To translate the graph of the function f(x) = 2x² + 5, 2 units down and 3 units left, we need to adjust the equation to incorporate these translations.
To move the graph 2 units down, we subtract 2 from the original function:
f(x) - 2 = 2x² + 5 - 2
To move the graph 3 units to the left, we substitute (x - 3) for x in the function:
f(x - 3) - 2 = 2(x - 3)² + 5 - 2
Simplifying further, we have:
f(x - 3) - 2 = 2(x² - 6x + 9) + 3
Expanding and combining like terms, we get:
f(x - 3) - 2 = 2x² - 12x + 18 + 3
Combining like terms again, we have:
f(x - 3) - 2 = 2x² - 12x + 21
Therefore, the function g(x) that represents the indicated translation of the graph of f is:
g(x) = 2x² - 12x + 21.