Find all possible values ( range) of the quadratic expression: `1+6x -x^2`, when `x in [-3,2]`.

Explanation

To find the range of the quadratic expression \( f(x) = 1 + 6x - x^2 \) for \( x \) in the interval \([-3, 2]\), we will follow these steps:

### Step 1: Rewrite the expression
The given expression can be rewritten as:
\[ f(x) = -x^2 + 6x + 1 \]

### Step 2: Identify the vertex of the quadratic
The vertex of a quadratic function in the form \( ax^2 + bx + c \) can be found using the formula:
\[ x = -\frac{b}{2a} \]
Here, \( a = -1 \) and \( b = 6 \). Thus,
\[ x = -\frac{6}{2 \cdot -1} = 3 \]

### Step 3: Determine if the vertex is within the interval
The vertex \( x = 3 \) is not within the interval \([-3, 2]\). Therefore, we need to evaluate the function at the endpoints of the interval.

### Step 4: Evaluate the function at the endpoints
1. Calculate \( f(-3) \):
\[ f(-3) = 1 + 6(-3) - (-3)^2 = 1 - 18 - 9 = -26 \]

2. Calculate \( f(2) \):
\[ f(2) = 1 + 6(2) - (2)^2 = 1 + 12 - 4 = 9 \]

### Step 5: Determine the range
Since the quadratic opens downwards (as the coefficient of \( x^2 \) is negative), the maximum value occurs at the right endpoint \( x = 2 \) and the minimum value occurs at the left endpoint \( x = -3 \).

Thus, the range of the function \( f(x) \) for \( x \in [-3, 2] \) is:
\[ [-26, 9] \]

### Final Answer
The range of the quadratic expression \( 1 + 6x - x^2 \) when \( x \) is in the interval \([-3, 2]\) is:
\[ [-26, 9] \]