Question
How does reflecting the function f(x) = (x+3)^2 over the x-axis affect the point (2,25)?
Answers
Answered by
GPT-4o mini
To reflect the function \( f(x) = (x + 3)^2 \) over the x-axis, we need to change the signs of the values of the function. Specifically, reflecting over the x-axis transforms \( f(x) \) into \( -f(x) \).
Given the point \( (2, 25) \), we can verify that this point is indeed on the curve of the original function by substituting \( x = 2 \):
\[
f(2) = (2 + 3)^2 = 5^2 = 25.
\]
Now, to find the new point after reflection over the x-axis, we take the same \( x \)-coordinate, which is \( 2 \), and negate the \( y \)-coordinate from the original point. The reflection will then be:
\[
(2, -f(2)) = (2, -25).
\]
Thus, the effect of reflecting the function over the x-axis results in the point \( (2, 25) \) becoming:
\[
(2, -25).
\]
Given the point \( (2, 25) \), we can verify that this point is indeed on the curve of the original function by substituting \( x = 2 \):
\[
f(2) = (2 + 3)^2 = 5^2 = 25.
\]
Now, to find the new point after reflection over the x-axis, we take the same \( x \)-coordinate, which is \( 2 \), and negate the \( y \)-coordinate from the original point. The reflection will then be:
\[
(2, -f(2)) = (2, -25).
\]
Thus, the effect of reflecting the function over the x-axis results in the point \( (2, 25) \) becoming:
\[
(2, -25).
\]
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