The problem asks to reflect the function $y = x^2 + 6x - 8$ across the line $y = x$, and then find the resulting function.

Step-by-step reflection process:

1. Initial Function: $y = x^2 + 6x - 8$ Here, the goal is to reflect the function across the line $y = x$. When reflecting a function across the line $y = x$, we swap the roles of $x$ and $y$, then solve for the new $y$.

2. Swap $x$ and $y$: $x = y^2 + 6y - 8$

3. Solve for $y$ (this becomes the new function):

   • Rearrange the equation into standard quadratic form: $y^2 + 6y - (x + 8) = 0$
   • Use the quadratic formula to solve for $y$: $y = \frac{-6 \pm \sqrt{6^2 - 4(1)(-(x + 8))}}{2(1)}$ Simplify: $y = \frac{-6 \pm \sqrt{36 + 4(x + 8)}}{2}$ $y = \frac{-6 \pm \sqrt{36 + 4x + 32}}{2}$ $y = \frac{-6 \pm \sqrt{68 + 4x}}{2}$ $y = \frac{-6 \pm \sqrt{4(x + 17)}}{2}$ $y = \frac{-6 \pm 2\sqrt{x + 17}}{2}$ $y = -3 \pm \sqrt{x + 17}$

4. Resulting Functions:

   • The two possible solutions are: $y = -3 + \sqrt{x + 17} \quad \text{or} \quad y = -3 - \sqrt{x + 17}$ These are the reflections of the original function across the line $y = x$.

Let me know if you need more details or have any questions!

Related Questions:

1. How do you reflect functions across the x-axis or y-axis?
2. What is the geometric interpretation of reflecting across $y = x$?
3. How can we graph the original and reflected functions for better visualization?
4. How would the reflection look if the original function was cubic?
5. Can reflection rules be applied to systems of equations?

Tip:

When reflecting a function over the line $y = x$, always swap the variables and solve for the new function.