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

Answer 1

To find the line of reflection for the given function f(x) = x² + 5x - 6 and its reflection f'(x) = -x² - 5x + 6, we can follow these steps:

1. Understand the reflection process: When a function is reflected over a line, the line of reflection is typically a horizontal line, which can be expressed as \(y = k\), where \(k\) is a constant. The value of \(k\) will be the average of the y-values of the original and reflected functions at corresponding x-values.

2. Find the midpoint: We can derive the line of reflection by computing the midpoint of the original function and reflected function.

   The midpoint \(M\) of \(y = f(x)\) and \(y = f'(x)\) is:

   \[ M = \frac{f(x) + f'(x)}{2} \]

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

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

3. The equation of the line of reflection: Therefore, the y-value of the line of reflection is \(0\).

Hence, the equation for the line of reflection is: \[ \boxed{y = 0} \]