To find the equation [tex]\( g(x) \)[/tex] for the graph of the function [tex]\( f(x) = x^2 \)[/tex] that is transformed as specified, follow these steps:
### Step-by-Step Solution:
1. Original Function:
Start with the given function:
[tex]\[
f(x) = x^2
\][/tex]
2. Shift 5 Units to the Left:
To shift the graph 5 units to the left, replace [tex]\( x \)[/tex] with [tex]\( x + 5 \)[/tex]:
[tex]\[
f(x) = (x + 5)^2
\][/tex]
3. Shift 2 Units Up:
To shift the graph 2 units up, add 2 to the function:
[tex]\[
f(x) = (x + 5)^2 + 2
\][/tex]
4. Vertically Shrink by [tex]\( \frac{1}{3} \)[/tex]:
To vertically shrink the function by a factor of [tex]\( \frac{1}{3} \)[/tex], multiply the entire function by [tex]\( \frac{1}{3} \)[/tex]:
[tex]\[
g(x) = \frac{1}{3} \left((x + 5)^2 + 2\right)
\][/tex]
### Final Transformed Function:
The new function [tex]\( g(x) \)[/tex] that represents the graph of [tex]\( f(x) = x^2 \)[/tex] shifted 5 units to the left, 2 units up, and vertically shrunk by [tex]\( \frac{1}{3} \)[/tex] is:
[tex]\[
g(x) = \frac{1}{3} \left((x + 5)^2 + 2\right)
\][/tex]
So, the final equation is:
[tex]\[
g(x) = \frac{1}{3} \left( (x + 5)^2 + 2 \right)
\][/tex]
