To find the new function [tex]\( g(x) \)[/tex] which is a translation of the given function [tex]\( f(x) = 2x^2 - 8 \)[/tex] down by 2 units, we need to follow these steps:
1. Understand the Translation:
Translating a function down by a fixed number of units means you subtract that number from the function. In this case, we subtract 2.
2. Apply the Translation:
The new function [tex]\( g(x) \)[/tex] is obtained by taking the original function [tex]\( f(x) \)[/tex] and subtracting 2. So we start with:
[tex]\[
g(x) = f(x) - 2
\][/tex]
3. Substitute [tex]\( f(x) \)[/tex] into the Equation:
We already know that [tex]\( f(x) = 2x^2 - 8 \)[/tex]. Substituting this into our equation for [tex]\( g(x) \)[/tex] gives:
[tex]\[
g(x) = (2x^2 - 8) - 2
\][/tex]
4. Simplify the Expression:
Combining the terms inside the parenthesis:
[tex]\[
g(x) = 2x^2 - 8 - 2
\][/tex]
Simplifying further:
[tex]\[
g(x) = 2x^2 - 10
\][/tex]
Therefore, the equation for [tex]\( g(x) \)[/tex] in its simplest form is:
[tex]\[
g(x) = 2x^2 - 10
\][/tex]
