To find the resulting function [tex]\( g(x) \)[/tex] after translating the graph of [tex]\( f(x) = x^2 \)[/tex] 3 units up and 1 unit left, let's follow these steps:
1. Translation 1 unit left:
When a function [tex]\( f(x) \)[/tex] is translated [tex]\( k \)[/tex] units to the left, you replace [tex]\( x \)[/tex] with [tex]\( x + k \)[/tex]. In this case, we replace [tex]\( x \)[/tex] with [tex]\( x + 1 \)[/tex].
Thus, the function becomes:
[tex]\[
f(x + 1) = (x + 1)^2
\][/tex]
2. Translation 3 units up:
When a function [tex]\( f(x) \)[/tex] is translated [tex]\( k \)[/tex] units up, you add [tex]\( k \)[/tex] to the function. In this case, we add 3 to the function:
[tex]\[
g(x) = (x + 1)^2 + 3
\][/tex]
So, the resulting function [tex]\( g(x) \)[/tex] after translating [tex]\( f(x) = x^2 \)[/tex] 1 unit to the left and 3 units up is:
[tex]\[
g(x) = (x + 1)^2 + 3
\][/tex]
