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The problem asks to find the range of the quadratic function \( f(x) = -2x^2 + 16x - 33 \) and express it in interval notation.

### Steps to find the range:

1. **Determine the vertex**:  
   Since the quadratic is in standard form \( f(x) = ax^2 + bx + c \), we use the vertex formula to find the x-coordinate of the vertex:
   \[
   x = -\frac{b}{2a}
   \]
   For \( f(x) = -2x^2 + 16x - 33 \), we have \( a = -2 \), \( b = 16 \), and \( c = -33 \).

   So,
   \[
   x = -\frac{16}{2(-2)} = \frac{16}{4} = 4
   \]

2. **Find the y-coordinate of the vertex**:  
   Substitute \( x = 4 \) into the quadratic function to get the corresponding y-value:
   \[
   f(4) = -2(4)^2 + 16(4) - 33
   \]
   \[
   f(4) = -2(16) + 64 - 33 = -32 + 64 - 33 = -1
   \]
   So, the vertex is \( (4, -1) \).

3. **Determine the range**:  
   Since the parabola opens downward (because \( a = -2 \) is negative), the vertex represents the maximum point of the function. Therefore, the maximum value of \( f(x) \) is \( -1 \), and the function continues decreasing to \( -\infty \) as \( x \) moves away from the vertex.

   The range is all y-values less than or equal to \( -1 \).

### Final answer in interval notation:
The range is \( (-\infty, -1] \).

Would you like more details on this explanation or have any questions?

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Here are 5 related questions:
1. How do you find the axis of symmetry for a quadratic function?
2. What is the significance of the sign of the leading coefficient in a quadratic equation?
3. How can the discriminant help determine the number of real roots of a quadratic equation?
4. How would the range change if the quadratic function opened upwards?
5. What are the properties of the graph of a quadratic function?

**Tip**: For any quadratic function, the vertex is a critical point to identify the function's range and direction.

Find the range of the quadratic function f(x) = -2x^2 + 16x - 33 and express it in interval notation.

Learn how to find the range of the quadratic function f(x) = -2x^2 + 16x - 33. This problem covers key algebraic concepts, including finding the vertex using the formula x = -b/(2a) and determining the function's maximum or minimum value. Suitable for students in Grades 9-12.

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