The graph of the function f(x) = x^2 will be translated 3 units up and 1 unit left. What is the resulting function g(x)?

The translation of the graph of a function is one where the graph is moved to a different location on the plane that does not include a change in shape or rotation

The resulting function from the translation of the function f(x) = x², 3 units up and 1 unit left, g(x) = x² + 2·x + 4

The process by which the above value for g(x) is found is presented as follows:

The given function f(x) = x²

The vertical translation given to the function = 3 units up

The horizontal translation given to the function  = 1 unit left

The required parameter;

To find the resulting function g(x) that has results from the given translations

Solution:

A translation of a function y = f(x) vertically, k units upwards is the function y = f(x) + k

A translation of a function y = f(x) horizontally, k, units left, is the function y = f(x + k)

Therefore, we get

g(x) = f(x + 1) + 3 = (x + 1)² + 3 = x² + 2·x + 4

g(x) = x² + 2·x + 4

Learn more about translation of functions here:

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 g(x) = (x + 1)² + 3

Step-by-step explanation:

The vertex form of a quadratic equation is: y = a(x - h)² + k    where

f(x) = x²

Given: 3 units up   -->   k = 3

           1 unit left    -->   h = -1

                     g(x) = (x + 1)² + 3