The graph of y=x−−√
 is translated 3 units to the left and 4 unitsdown. What is the equation of the graph that results from this translation?

To find the equation of the graph after a translation, we apply the transformations to the original equation.

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

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

### Step 2: Translation 4 Units Down
- When a graph is translated vertically downward by 4 units, we subtract 4 from the entire function.
- The equation now becomes:
\[ y = \sqrt{x + 3} - 4 \]

So, the equation of the graph after the translation is:
\[
\boxed{y = \sqrt{x + 3} - 4}
\]

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 5 units to the right?
2. What is the effect of translating the graph 2 units up?
3. How does horizontal stretching affect the equation of a graph?
4. What happens to the equation if the graph is reflected over the x-axis?
5. Can you describe the transformation if the graph is moved 3 units to the right and 2 units up?
6. What is the domain of the function after the translation?
7. How would you translate the graph of \( y = \sqrt{x + 3} - 4 \) back to its original position?
8. How do combined transformations (translation and reflection) affect the final equation?

**Tip:** When performing multiple transformations, always apply horizontal changes (translations, stretches) before vertical changes (translations, stretches) for clarity and accuracy.

Understand how to translate the square root function y = sqrt(x) 3 units to the left and 4 units down. Learn step-by-step with detailed explanations and examples.

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