For f(x)=x−6−−−−√, what is the vertex of f(x)after being horizontally dilated by a factor of 2?

Given the function \( f(x) = \sqrt{x - 6} \), we need to determine the vertex after applying a horizontal dilation by a factor of 2.

### Step-by-Step Process:

1. **Original Function**: 
   The function is \( f(x) = \sqrt{x - 6} \). This is a square root function, which typically has the form \( f(x) = \sqrt{x - h} \), where \( h \) determines the horizontal shift. For the given function, \( h = 6 \), so the vertex of the original function is \( (6, 0) \).

2. **Horizontal Dilation**:
   A horizontal dilation by a factor of 2 means that the input to the function is multiplied by \( \frac{1}{2} \). Therefore, the dilated function becomes:
   \[
   f_{\text{dilated}}(x) = \sqrt{\frac{x}{2} - 6}
   \]
   or equivalently:
   \[
   f_{\text{dilated}}(x) = \sqrt{\frac{1}{2}(x - 12)}
   \]
   Thus, the vertex of the new function occurs when the expression inside the square root is zero, i.e.:
   \[
   \frac{x}{2} - 6 = 0 \quad \Rightarrow \quad x = 12
   \]
   
3. **New Vertex**: 
   After the horizontal dilation, the vertex shifts to \( (12, 0) \).

### Final Answer:
The vertex of the function after being horizontally dilated by a factor of 2 is \( (12, 0) \).

---

Would you like more details, or do you have any further questions?

Here are some related questions to deepen understanding:
1. How would the function change with a vertical dilation instead of horizontal?
2. What happens to the domain of \( f(x) = \sqrt{x - 6} \) after the dilation?
3. How would you find the range of the dilated function?
4. What effect would a reflection across the y-axis have on the vertex?
5. How does shifting the function vertically affect the vertex?

**Tip**: Horizontal dilations affect the x-values, scaling them by the reciprocal of the dilation factor.

For f(x)=√(x−6), what is the vertex of f(x) after being horizontally dilated by a factor of 2?

Learn how to find the vertex of the square root function f(x) = √(x−6) after a horizontal dilation by a factor of 2. This problem involves function transformations, including horizontal dilation and vertex shifting, and is suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation