for f(x)=x−6−−−−√, what is the vertex of f(x) after being horizontally dilated by a factor of 2?write your answer as an ordered pair.

The given function is:

\[
f(x) = \sqrt{x - 6}
\]

To horizontally dilate this function by a factor of 2, we replace \( x \) with \( \frac{x}{2} \). The new function becomes:

\[
f(x) = \sqrt{\frac{x}{2} - 6}
\]

The vertex of the original function \( f(x) = \sqrt{x - 6} \) occurs where the expression inside the square root equals zero, i.e., \( x - 6 = 0 \), so \( x = 6 \). Therefore, the vertex of the original function is \( (6, 0) \).

For the horizontally dilated function \( f(x) = \sqrt{\frac{x}{2} - 6} \), we set the expression inside the square root to zero:

\[
\frac{x}{2} - 6 = 0
\]

Solving for \( x \):

\[
\frac{x}{2} = 6 \quad \Rightarrow \quad x = 12
\]

At this point, the value of \( f(x) \) is \( 0 \), so the vertex is at \( (12, 0) \).

Thus, the vertex of the horizontally dilated function is \( \boxed{(12, 0)} \).

Would you like more details or have any questions?

Here are 5 related questions to deepen understanding:

1. How do vertical dilations affect the vertex of a function?
2. What is the domain of the horizontally dilated function \( f(x) = \sqrt{\frac{x}{2} - 6} \)?
3. How would the vertex change if we applied a vertical dilation instead of a horizontal one?
4. Can you explain why the horizontal dilation caused the vertex to shift from \( (6, 0) \) to \( (12, 0) \)?
5. What happens to the graph of \( f(x) = \sqrt{x - 6} \) if it is reflected horizontally?

**Tip:** Remember, horizontal transformations (like dilations and translations) affect the \( x \)-coordinate of the vertex, while vertical transformations affect the \( y \)-coordinate.

For f(x)=√(x−6), what is the vertex of f(x) after being horizontally dilated by a factor of 2? Write your answer as an ordered pair.

This problem explores the transformation of the square root function f(x)=√(x−6) after a horizontal dilation by a factor of 2. Understand how to apply the dilation and find the vertex of the new function, with a step-by-step solution. Suitable for high school students (Grades 9-12).

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