To graph the function \( f(x) = (x + 1)^2 \) and its horizontal reflection \( f(-x) \), let's first determine each function.
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Original Function: \[ f(x) = (x + 1)^2 \] This is a parabola that opens upwards, with its vertex at the point \( (-1, 0) \).
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Horizontal Reflection: The horizontal reflection of a function \( f(x) \) is obtained by replacing \( x \) with \( -x \). So, we have: \[ f(-x) = (-x + 1)^2 = (1 - x)^2 \] This is also a parabola opening upwards, with its vertex at the point \( (1, 0) \).
Now, let's find the point that corresponds to the point \( (-4, 9) \) in the reflected function \( f(-x) \).
To reflect the point \( (-4, 9) \) over the y-axis:
- The x-coordinate will change sign, so the reflected point will be \( (4, 9) \).
Now, we need to check if this reflected point is actually on \( f(-x) \):
- Substitute \( x = 4 \) into \( f(-x) \): \[ f(-4) = (1 - (-4))^2 = (1 + 4)^2 = 5^2 = 25 \]
Thus, the point \( (4, 9) \) does not lie on \( f(-x) \), as \( f(-4) = 25 \).
In conclusion, the reflection of the point \( (-4, 9) \) in the context of the reflected function \( f(-x) \) is: \[ \text{The point (-4, 9) reflects to (4, 9) in } f(-x). \]
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