Sure, let's transform the quadratic function step-by-step.
Given:
[tex]\[ f(x) = \frac{1}{2} x^2 + 4 \][/tex]
We need to find the function [tex]\( g(x) \)[/tex] which is a vertical stretch of [tex]\( f(x) \)[/tex] by a factor of 4.
### Step-by-Step Solution to Find [tex]\( g(x) \)[/tex]:
1. Identify the Transformation:
The transformation required is a vertical stretch by a factor of 4. That means the output of [tex]\( f(x) \)[/tex] is multiplied by 4.
2. Apply the Stretch:
To find [tex]\( g(x) \)[/tex], multiply [tex]\( f(x) \)[/tex] by 4:
[tex]\[ g(x) = 4 \cdot f(x) \][/tex]
3. Substitute [tex]\( f(x) \)[/tex]:
Substitute the given [tex]\( f(x) \)[/tex] into this expression:
[tex]\[ g(x) = 4 \left( \frac{1}{2} x^2 + 4 \right) \][/tex]
4. Distribute the Factor:
Distribute the factor of 4 to both terms inside the parentheses:
[tex]\[ g(x) = 4 \cdot \frac{1}{2} x^2 + 4 \cdot 4 \][/tex]
[tex]\[ g(x) = 2 x^2 + 16 \][/tex]
### Conclusion
The equation of [tex]\( g(x) \)[/tex] after the vertical stretch is:
[tex]\[
g(x) = 2x^2 + 16
\][/tex]
So, the final answer is:
[tex]\[
g(x) = 2x^2 + 16
\][/tex]
