The function f(x)=x2+5x−6

Question

𝑓
(
𝑥
)
=
𝑥
2
+
5
𝑥
−
6
is reflected over a given line and results in the equation f′(x)=−x2−5x+6
𝑓
′
(
𝑥
)
=
−
𝑥
2
−
5
𝑥
+
6
. Identify the equation for the line of reflection. (1 point)
The line of reflection is y=
𝑦
=

Answer 1

To find the line of reflection for the given functions \( f(x) = x^2 + 5x - 6 \) and \( f'(x) = -x^2 - 5x + 6 \), we can use the property that the line of reflection will be the average of the two functions at any point where they intersect.

Given the symmetric nature of reflection, we can find the midpoint of \( f(x) \) and \( f'(x) \):

1. Calculate the average of the two functions: \[ \text{Midpoint} = \frac{f(x) + f'(x)}{2} \]

   Substitute \( f(x) \) and \( f'(x) \): \[ \text{Midpoint} = \frac{(x^2 + 5x - 6) + (-x^2 - 5x + 6)}{2} \]

   Simplifying: \[ = \frac{x^2 + 5x - 6 - x^2 - 5x + 6}{2} = \frac{0}{2} = 0 \]

2. Therefore, the y-coordinate of the midpoint is 0, and since the reflection is over a horizontal line, the line of reflection is simply \( y = 0 \).

Thus, the equation for the line of reflection is: \[ y = 0 \]