what is the range of the function f(x)=(x-4)^2+1

To find the range of the function \( f(x) = (x - 4)^2 + 1 \), let's analyze it step by step:

### 1. **Form of the function:**
   The function is a quadratic function, written as \( f(x) = (x - 4)^2 + 1 \). This is a transformation of the basic quadratic function \( y = x^2 \), which is a parabola opening upwards.

### 2. **Vertex:**
   The function \( f(x) = (x - 4)^2 + 1 \) is in vertex form, \( f(x) = (x - h)^2 + k \), where \( (h, k) \) is the vertex of the parabola. 
   - In this case, \( h = 4 \) and \( k = 1 \), so the vertex is at \( (4, 1) \).

### 3. **Behavior of the function:**
   Since the parabola opens upwards (because the coefficient of \( (x - 4)^2 \) is positive), the minimum value of \( f(x) \) occurs at the vertex. At the vertex, the function takes the value \( f(4) = (4 - 4)^2 + 1 = 0 + 1 = 1 \).

### 4. **Range:**
   The minimum value of the function is \( 1 \) (at \( x = 4 \)), and as \( x \) moves away from 4 in either direction, the value of \( f(x) \) increases. Thus, the range of the function is all values greater than or equal to 1.

### Final Answer:
The range of the function is \( [1, \infty) \).

Would you like more details or have any questions?

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Here are 5 related questions to further explore the concept:
1. What is the domain of the function \( f(x) = (x - 4)^2 + 1 \)?
2. How would the range change if the function were \( f(x) = -(x - 4)^2 + 1 \)?
3. Can you find the x-intercepts of the function \( f(x) = (x - 4)^2 + 1 \)?
4. What is the axis of symmetry for the function \( f(x) = (x - 4)^2 + 1 \)?
5. How would shifting the vertex affect the range of this quadratic function?

**Tip:** In vertex form \( f(x) = a(x-h)^2 + k \), the value of \( k \) always determines the minimum or maximum value depending on the direction the parabola opens.

Discover how to find the range of the quadratic function f(x) = (x-4)^2 + 1. This solution explains the concept of vertex form and how the minimum value of the function determines the range. Ideal for students in Grades 9-11 studying quadratic functions.

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