Write an equation for a function that has the graph with the shape of yequals=StartRoot x EndRootx​, but reflected across the​ y-axis and shifted rightright 88 units and upup 55 

To write an equation for the function that follows these transformations, we'll start with the base function:

\[
y = \sqrt{x}
\]

Now, let's apply the specified transformations:

1. **Reflection across the y-axis**: To reflect a function across the y-axis, we replace \( x \) with \( -x \). This gives:
   \[
   y = \sqrt{-x}
   \]

2. **Shift 8 units to the right**: To shift the graph to the right by 8 units, we replace \( x \) with \( x - 8 \). Applying this to the current function:
   \[
   y = \sqrt{-(x - 8)}
   \]
   Simplifying this:
   \[
   y = \sqrt{8 - x}
   \]

3. **Shift 5 units up**: To shift the graph up by 5 units, we add 5 to the function:
   \[
   y = \sqrt{8 - x} + 5
   \]

Thus, the final equation for the function is:
\[
y = \sqrt{8 - x} + 5
\]

Would you like any further details or clarifications on this? Here are 5 questions to explore further:

1. How do you reflect a function across the x-axis?
2. What happens if you shift a function left instead of right?
3. What if you combined vertical and horizontal stretches with these transformations?
4. How would this function change if you reflected it across the x-axis instead?
5. How can you identify these transformations visually on a graph?

**Tip:** When reflecting a function across the y-axis, always replace \( x \) with \( -x \) inside the function.

Write an equation for a function that has the graph with the shape of y = √x, but reflected across the y-axis and shifted right 8 units and up 5.

Explore how to transform the square root function y = √x by reflecting it across the y-axis, shifting it 8 units to the right, and 5 units up. This detailed solution demonstrates key function transformation concepts, suitable for students in Grades 9-11.

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