To determine the function [tex]\( g(x) \)[/tex] when the function [tex]\( f(x) = x^2 + 5x - 6 \)[/tex] is shifted 4 units to the left, follow these steps:
### Step 1: Understanding the Shift
When a function [tex]\( f(x) \)[/tex] is shifted [tex]\( k \)[/tex] units to the left, it means that every instance of [tex]\( x \)[/tex] in the function is replaced by [tex]\( (x + k) \)[/tex].
### Step 2: Applying the Shift
Here, the function is shifted 4 units to the left, so we replace every [tex]\( x \)[/tex] in [tex]\( f(x) \)[/tex] with [tex]\( (x + 4) \)[/tex].
Therefore, the new function [tex]\( g(x) \)[/tex] will be:
[tex]\[
g(x) = f(x + 4)
\][/tex]
### Step 3: Substitute [tex]\( x + 4 \)[/tex] into [tex]\( f(x) \)[/tex]
Given [tex]\( f(x) = x^2 + 5x - 6 \)[/tex], substituting [tex]\( x + 4 \)[/tex] for [tex]\( x \)[/tex] gives us:
[tex]\[
g(x) = (x + 4)^2 + 5(x + 4) - 6
\][/tex]
### Step 4: Simplify [tex]\( g(x) \)[/tex]
Expand and simplify the expression:
1. Expand [tex]\( (x + 4)^2 \)[/tex]:
[tex]\[
(x + 4)^2 = x^2 + 8x + 16
\][/tex]
2. Expand [tex]\( 5(x + 4) \)[/tex]:
[tex]\[
5(x + 4) = 5x + 20
\][/tex]
3. Combine these with the constant [tex]\(-6\)[/tex]:
[tex]\[
g(x) = x^2 + 8x + 16 + 5x + 20 - 6
\][/tex]
4. Combine like terms:
[tex]\[
g(x) = x^2 + 13x + 30
\][/tex]
### Conclusion
The simplified form of [tex]\( g(x) \)[/tex] after shifting [tex]\( f(x) = x^2 + 5x - 6 \)[/tex] 4 units to the left is:
[tex]\[
g(x) = x^2 + 13x + 30
\][/tex]
Thus, the correct answer is:
C. [tex]\( g(x) = (x + 4)^2 + 5(x + 4) - 6 \)[/tex]
Notice that this represents the form before simplification. The simplification leads to the expression:
[tex]\[
g(x) = x^2 + 13x + 30
\][/tex]
The final simplified version confirms the calculated transformation.
