Sure, let’s break down how each transformation affects the given function [tex]\( f(x) = x^2 \)[/tex] step-by-step.
### Step-by-Step Solution:
1. Translation 2 units up:
- When you translate a function [tex]\( f(x) \)[/tex] vertically by adding a constant [tex]\( k \)[/tex], the new function becomes [tex]\( f(x) + k \)[/tex].
- For [tex]\( f(x) = x^2 \)[/tex] and a translation 2 units up, the new function [tex]\( g(x) \)[/tex] becomes:
[tex]\[
g(x) = f(x) + 2 = x^2 + 2
\][/tex]
2. Reflection in the x-axis:
- Reflecting a function [tex]\( f(x) \)[/tex] in the x-axis involves multiplying the function by -1.
- Thus, the function after reflecting [tex]\( x^2 + 2 \)[/tex] in the x-axis is:
[tex]\[
g(x) = -(x^2 + 2) = -x^2 - 2
\][/tex]
3. Vertical stretch by a factor of 6:
- To vertically stretch a function by a factor of [tex]\( k \)[/tex], you multiply the entire function by [tex]\( k \)[/tex].
- For a vertical stretch by a factor of 6, the function [tex]\( -x^2 - 2 \)[/tex] changes to:
[tex]\[
g(x) = 6 \cdot (-x^2 - 2) = -6x^2 - 12
\][/tex]
Therefore, after applying all the given transformations (translation 2 units up, reflection in the x-axis, and vertical stretch by a factor of 6) to the original function [tex]\( f(x) = x^2 \)[/tex], the resulting function [tex]\( g(x) \)[/tex] is:
[tex]\[
g(x) = -6x^2 - 12
\][/tex]
