What is the range of this quadratic function?

y

=

–

x

2

–

2

x

+

8

To find the range of the quadratic function \( y = -x^2 - 2x + 8 \), let's go through the steps:

### 1. Identify the vertex of the parabola:
The function is in the form \( y = ax^2 + bx + c \), where \( a = -1 \), \( b = -2 \), and \( c = 8 \).

The x-coordinate of the vertex is calculated using the formula:
\[
x = \frac{-b}{2a}
\]
Substituting the values of \( a \) and \( b \):
\[
x = \frac{-(-2)}{2(-1)} = \frac{2}{-2} = -1
\]

### 2. Find the y-coordinate of the vertex:
Substitute \( x = -1 \) back into the equation to find \( y \):
\[
y = -(-1)^2 - 2(-1) + 8
\]
\[
y = -1 + 2 + 8 = 9
\]

### 3. Determine the range:
Since the coefficient of \( x^2 \) (which is \( a = -1 \)) is negative, the parabola opens downward. This means the vertex is the maximum point.

Therefore, the range of the function is:
\[
(-\infty, 9]
\]

Would you like more details or have any questions?

### 5 Related Questions:
1. How do you find the x-intercepts of a quadratic function?
2. What is the significance of the vertex in determining the maximum or minimum value?
3. How can you convert a quadratic function to vertex form?
4. How does the value of \( c \) in the quadratic function affect the graph?
5. What is the relationship between the axis of symmetry and the vertex?

### Tip:
The range of a quadratic function depends on whether the parabola opens upward or downward. The vertex provides the highest or lowest point, which helps determine the range.

Discover how to find the range of the quadratic function y = -x^2 - 2x + 8 using vertex analysis. This problem explores key concepts in quadratic functions, including vertex calculation and determining function range. Suitable for Grades 10-12.

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