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 $y=f^{\prime }(x)$ is:

   $$M=\frac{f(x)+f^{\prime }(x)}{2}$$

   Substitute $f(x)$ and $f^{\prime }(x)$: $$M=\frac{(x^2+5x-6)+(-x^2-5x+6)}{2}$$

   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: $$\boxed{{\displaystyle y=0}}$$