4x^{2}-32x+66. What is the range of f(x)? On what interval is the function increasing and decreasing?

The function given is \( f(x) = 4x^2 - 32x + 66 \), which is a quadratic function of the form \( f(x) = ax^2 + bx + c \), where \( a = 4 \), \( b = -32 \), and \( c = 66 \).

### Step 1: Determine the Range
Since this is a parabola that opens upwards (because \( a > 0 \)), the minimum value of the function occurs at the vertex.

To find the vertex of the parabola, we use the formula for the \( x \)-coordinate of the vertex:

\[
x_{\text{vertex}} = \frac{-b}{2a} = \frac{-(-32)}{2(4)} = \frac{32}{8} = 4
\]

Next, substitute \( x = 4 \) into the function to find the \( y \)-coordinate of the vertex, which gives the minimum value of \( f(x) \):

\[
f(4) = 4(4)^2 - 32(4) + 66 = 4(16) - 128 + 66 = 64 - 128 + 66 = 2
\]

So, the vertex is at \( (4, 2) \). Since the parabola opens upwards, the minimum value is \( 2 \), and the range of the function is:

\[
\boxed{[2, \infty)}
\]

### Step 2: Determine Intervals of Increase and Decrease
To determine where the function is increasing or decreasing, we look at the vertex and the direction the parabola opens:

- **Decreasing**: The function decreases on the interval \( (-\infty, 4] \).
- **Increasing**: The function increases on the interval \( [4, \infty) \).

### Final Answer:
- **Range**: \( [2, \infty) \)
- **Interval of decrease**: \( (-\infty, 4] \)
- **Interval of increase**: \( [4, \infty) \)

Would you like further explanation on any step, or have any other questions?

### Related Questions:
1. How do you find the vertex of a general quadratic function?
2. Why does a parabola open upwards or downwards depending on the coefficient of \(x^2\)?
3. Can you explain how to derive the formula for the range of a quadratic function?
4. What would change in the range if the parabola opened downwards?
5. How can you graph a quadratic function based on its vertex and direction?

### Tip:
The vertex form of a quadratic function \( f(x) = a(x - h)^2 + k \) makes it easier to identify the vertex and determine the range directly.

4x^2 - 32x + 66. What is the range of f(x)? On what interval is the function increasing and decreasing?

Explore how to determine the range and intervals of increase and decrease for the quadratic function f(x) = 4x^2 - 32x + 66. This guide covers algebraic concepts, including vertex calculation and analyzing quadratic functions, suitable for high school students in Grades 9-12.

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