Answer :
To solve the problem of determining which equation represents the new graph when [tex]\( f(x) = x \)[/tex] is shifted up 9 units, let's break it down step-by-step:
1. Understanding the Shift:
- When a graph is shifted vertically, we add or subtract a constant to the function.
- If the graph is shifted up by a certain number of units, we add that number to the original function.
2. Original Function:
- The original function given is [tex]\( f(x) = x \)[/tex].
3. Shifting Up:
- To shift the graph of [tex]\( f(x) = x \)[/tex] up by 9 units, we add 9 to the function.
- Mathematically, this means: [tex]\( g(x) = f(x) + 9 \)[/tex].
4. Applying the Shift:
- Substitute [tex]\( f(x) \)[/tex] into the new function: [tex]\( g(x) = x + 9 \)[/tex].
- This matches the form given in choice A: [tex]\( g(x) = f(x) + 9 \)[/tex].
5. Verifying Other Choices:
- Choice B: [tex]\( g(x) = f(x) - 9 \)[/tex] would represent a downward shift by 9 units, which is incorrect.
- Choice C: [tex]\( g(x) = 9 f(x) \)[/tex] would represent a vertical stretch, not a shift.
- Choice D: [tex]\( g(x) = 9 - f(x) \)[/tex] would flip the graph and shift it, which is also incorrect.
Thus, the correct equation for the new graph after shifting [tex]\( f(x) = x \)[/tex] up by 9 units is:
[tex]\[ \boxed{g(x) = f(x) + 9} \][/tex]
The corresponding correct answer is:
[tex]\[ 1 \][/tex]
1. Understanding the Shift:
- When a graph is shifted vertically, we add or subtract a constant to the function.
- If the graph is shifted up by a certain number of units, we add that number to the original function.
2. Original Function:
- The original function given is [tex]\( f(x) = x \)[/tex].
3. Shifting Up:
- To shift the graph of [tex]\( f(x) = x \)[/tex] up by 9 units, we add 9 to the function.
- Mathematically, this means: [tex]\( g(x) = f(x) + 9 \)[/tex].
4. Applying the Shift:
- Substitute [tex]\( f(x) \)[/tex] into the new function: [tex]\( g(x) = x + 9 \)[/tex].
- This matches the form given in choice A: [tex]\( g(x) = f(x) + 9 \)[/tex].
5. Verifying Other Choices:
- Choice B: [tex]\( g(x) = f(x) - 9 \)[/tex] would represent a downward shift by 9 units, which is incorrect.
- Choice C: [tex]\( g(x) = 9 f(x) \)[/tex] would represent a vertical stretch, not a shift.
- Choice D: [tex]\( g(x) = 9 - f(x) \)[/tex] would flip the graph and shift it, which is also incorrect.
Thus, the correct equation for the new graph after shifting [tex]\( f(x) = x \)[/tex] up by 9 units is:
[tex]\[ \boxed{g(x) = f(x) + 9} \][/tex]
The corresponding correct answer is:
[tex]\[ 1 \][/tex]
