Question

Graph the function f(x)=(x+1)2
𝑓
(
𝑥
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=
(
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+
1
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2
and the function of its horizontal reflection. What is the position of the point (−4,9)
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−
4
,
9
)
in the reflected function?(1 point)
The point (−4,9)
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−
4
,
9
)
reflects to () in f(−x)
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−
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.

Answer 1

To analyze the function \( f(x) = (x + 1)^2 \) and its horizontal reflection \( f(-x) \), let's first rewrite and evaluate both functions.

### Original Function
1. The function is \( f(x) = (x + 1)^2 \).
2. To graph \( f(x) \), notice the vertex of this parabola is at \( (-1, 0) \) and it opens upwards.
3. Some key points to plot:
- \( f(-1) = 0 \) (vertex)
- \( f(0) = (0 + 1)^2 = 1 \)
- \( f(-2) = (-2 + 1)^2 = 1 \)
- \( f(-3) = (-3 + 1)^2 = 4 \)
- \( f(-4) = (-4 + 1)^2 = 9 \) (point (−4,9))

### Reflected Function
1. The reflected function is given by \( f(-x) = (-x + 1)^2 = (1 - x)^2 \).
2. To graph \( f(-x) \), you can identify its vertex and plot additional points:
- The vertex of \( f(-x) \) is at \( (1, 0) \).
- Some key points include:
- \( f(-(-1)) = f(1) = (1 - 1)^2 = 0 \)
- \( f(-0) = f(0) = (1 - 0)^2 = 1 \)
- \( f(-2) = f(2) = (1 - 2)^2 = 1 \)
- \( f(-3) = f(3) = (1 - 3)^2 = 4 \)
- \( f(-4) = f(4) = (1 - 4)^2 = 9 \) (calculation given point (−4,9))

#### Position of point \( (-4, 9) \)
We need to find the output of the reflection of the input \( -4 \) in the new function \( f(-x) \):
- The point \( (-4, 9) \) in \( f(x) \) reflects to \( (4, 9) \) in \( f(-x) \).

Thus, the reflection of the point \( (-4, 9) \) in \( f(-x) \) is:
\[
\boxed{(4, 9)}
\]