To determine which point lies on the graph of the transformed function, we need to first understand the transformations applied to the original function [tex]\( f(x) = |x| \)[/tex].
### Step-by-Step Solution:
1. Original Function:
The original function given is [tex]\( f(x) = |x| \)[/tex].
2. Translation Left by 2 Units:
Translating a function [tex]\( f(x) \)[/tex] left by 2 units changes the function to [tex]\( f(x + 2) \)[/tex].
3. Translation Up by 1 Unit:
Translating a function [tex]\( f(x) \)[/tex] up by 1 unit changes the function to [tex]\( f(x) + 1 \)[/tex].
4. Combining the Transformations:
Apply the transformations one after the other:
- First, translate left by 2 units to get [tex]\( f(x + 2) \)[/tex].
- Then, translate up by 1 unit, which gives us [tex]\( f(x + 2) + 1 \)[/tex].
Thus, our transformed function becomes:
[tex]\[
g(x) = |x + 2| + 1
\][/tex]
5. Evaluating Each Point:
Now, we need to check which of the given points satisfies the transformed function [tex]\( g(x) = |x + 2| + 1 \)[/tex].
- For point [tex]\((-4, 2)\)[/tex]:
[tex]\[
g(-4) = |-4 + 2| + 1 = |-2| + 1 = 2 + 1 = 3 \quad (\text{not equal to } 2)
\][/tex]
- For point [tex]\((-3, 1)\)[/tex]:
[tex]\[
g(-3) = |-3 + 2| + 1 = |-1| + 1 = 1 + 1 = 2 \quad (\text{not equal to } 1)
\][/tex]
- For point [tex]\((-2, 5)\)[/tex]:
[tex]\[
g(-2) = |-2 + 2| + 1 = |0| + 1 = 0 + 1 = 1 \quad (\text{not equal to } 5)
\][/tex]
- For point [tex]\((-1, 2)\)[/tex]:
[tex]\[
g(-1) = |-1 + 2| + 1 = |1| + 1 = 1 + 1 = 2 \quad (\text{equal to } 2)
\][/tex]
6. Conclusion:
Only the point [tex]\((-1, 2)\)[/tex] satisfies the equation [tex]\( g(x) = |x + 2| + 1 \)[/tex]. Therefore, the point that lies on the new graph is:
[tex]\[
(-1, 2)
\][/tex]
