maximum or minimum and range for f(x)=x^2-8x+12

To analyze the function \( f(x) = x^2 - 8x + 12 \), we'll find the following:

1. **Vertex** (to determine whether the function has a maximum or minimum, and the value at that point).
2. **Range** (the set of possible values of \( f(x) \)).

### 1. Maximum or Minimum
The function \( f(x) = x^2 - 8x + 12 \) is a quadratic function in the form \( ax^2 + bx + c \), where:
- \( a = 1 \) (positive, indicating the parabola opens upwards).
- \( b = -8 \)
- \( c = 12 \)

Since \( a = 1 > 0 \), the parabola opens upwards, meaning the function has a **minimum**.

The vertex occurs at \( x = -\frac{b}{2a} \). For this function:
\[
x = -\frac{-8}{2(1)} = \frac{8}{2} = 4
\]
Now, substitute \( x = 4 \) back into the function to find the minimum value:
\[
f(4) = (4)^2 - 8(4) + 12 = 16 - 32 + 12 = -4
\]
So, the minimum value of \( f(x) \) is \( -4 \), and it occurs at \( x = 4 \).

### 2. Range
Since the function has a minimum value at \( x = 4 \), and it opens upwards, the range is all values of \( f(x) \) greater than or equal to \( -4 \). Thus, the range is:
\[
\text{Range} = [-4, \infty)
\]

### Summary:
- **Minimum value**: \( -4 \) at \( x = 4 \)
- **Range**: \( [-4, \infty) \)

Would you like further details or have any questions?

Here are 5 related questions:
1. How do you determine if a quadratic function opens upwards or downwards?
2. What is the formula for finding the vertex of a quadratic function?
3. How do you find the axis of symmetry in a quadratic function?
4. What does the discriminant tell us about the roots of a quadratic function?
5. Can you describe how to complete the square for the function \( f(x) = x^2 - 8x + 12 \)?

**Tip**: Always check the sign of \( a \) in a quadratic function to quickly determine if it has a maximum (if \( a < 0 \)) or a minimum (if \( a > 0 \)).

maximum or minimum and range for f(x) = x^2 - 8x + 12

Learn how to find the minimum value and range for the quadratic function f(x) = x^2 - 8x + 12. This detailed solution covers the vertex formula, algebraic techniques, and how to determine whether the function has a maximum or minimum. Ideal for students in Grades 9-10.

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