To graph the function \( f(x) = (x + 1)^2 \) and its horizontal reflection, we first find the horizontal reflection of the function.
The horizontal reflection of \( f(x) \) is given by \( f(-x) \). To find \( f(-x) \), we substitute \(-x\) for \(x\) in the original function:
\[ f(-x) = (-x + 1)^2 = (1 - x)^2 \]
Now, let's evaluate the point \( (-4, 9) \) in the context of the reflected function \( f(-x) \).
Step 1: Determine \( f(-x) \) at \( x = -4 \)
We need to determine if \( (-4, 9) \) lies on \( f(x) \) first:
\[ f(-4) = ((-4) + 1)^2 = (-3)^2 = 9 \]
So, indeed, the point \( (-4, 9) \) lies on \( f(x) \).
Step 2: Find the corresponding point in \( f(-x) \)
Now to find the reflected position of the point \((-4, 9)\) in \( f(-x) \), we need to substitute \( x = 4 \) into \( f(-4) \):
To find the point in \( f(-x) \) corresponding to \( (-4, 9) \):
The original point is \((-4, 9)\), and because the function has a horizontal reflection, the reflected point will be:
\[ (4, f(-4)) \] We already calculated \( f(-4) \) which is \( 9 \).
Thus, the reflected point is \( (4, 9) \).
Conclusion
The position of the point \((-4, 9)\) in the reflected function \( f(-x) \) would be:
\[ (4, 9) \]
So you fill in the blanks as follows:
The point \((-4,9)\) reflects to \( (4, 9) \) in \( f(-x) \).