The graph of y=x−−√
 is translated 3 units to the right and slid 5 units up, what would be the resulting equation?

To find the resulting equation after translating the graph, we'll apply the appropriate transformations to the original equation.

The original equation is:
\[ y = \sqrt{x} \]

### Step 1: Translation 3 Units to the Right
- When a graph is translated horizontally to the right by 3 units, we replace \( x \) with \( x - 3 \).
- The equation becomes:
\[ y = \sqrt{x - 3} \]

### Step 2: Translation 5 Units Up
- When a graph is translated vertically upward by 5 units, we add 5 to the entire function.
- The equation now becomes:
\[ y = \sqrt{x - 3} + 5 \]

So, the equation of the graph after the translation is:
\[
\boxed{y = \sqrt{x - 3} + 5}
\]

Would you like more details or have any questions?

Here are 8 related questions to expand your understanding:
1. How would the equation change if the graph was translated 2 units to the left?
2. What is the effect of translating the graph 4 units down?
3. How does vertical stretching affect the equation of a graph?
4. What happens to the equation if the graph is reflected over the y-axis?
5. Can you describe the transformation if the graph is moved 6 units to the left and 3 units up?
6. What is the domain of the function after this translation?
7. How would you translate the graph of \( y = \sqrt{x - 3} + 5 \) back to its original position?
8. How do multiple transformations (such as translation and reflection) impact the final graph?

**Tip:** Always verify the order of transformations—horizontal shifts should be considered first, followed by vertical shifts—to avoid mistakes in the final equation.

Explore how to transform the square root function y = sqrt(x) by translating it 3 units to the right and 5 units up. Learn the step-by-step process to derive the resulting equation y = sqrt(x - 3) + 5. Suitable for high school students studying functions and transformations.

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