what is the range. function is g(x)=-(x+6)^2+1

To find the range of the function \( g(x) = -(x + 6)^2 + 1 \), let's analyze it step-by-step.

### Step 1: Identify the type of function

The function \( g(x) = -(x + 6)^2 + 1 \) is a quadratic function in the form of \( g(x) = a(x - h)^2 + k \), where:
- \( a = -1 \) (negative, indicating a downward parabola),
- \( h = -6 \) (horizontal shift),
- \( k = 1 \) (vertical shift).

### Step 2: Determine the vertex

The vertex of a parabola \( y = a(x - h)^2 + k \) is \((h, k)\). Here, the vertex is:
\[
(-6, 1)
\]

Since \( a = -1 \), the parabola opens downward, making the vertex a maximum point.

### Step 3: Find the range

- The maximum value of \( g(x) \) occurs at the vertex, which is \( 1 \).
- Since the parabola opens downward, \( g(x) \) can take any value less than or equal to 1.

Therefore, the range of \( g(x) \) is:
\[
\boxed{(-\infty, 1]}
\]

Would you like more details, or do you have any questions? 

### Related Questions:
1. What is the domain of \( g(x) = -(x + 6)^2 + 1 \)?
2. How would the range change if the function were \( g(x) = (x + 6)^2 + 1 \)?
3. What are the coordinates of the vertex of a function in the form \( a(x - h)^2 + k \)?
4. How does the sign of \( a \) affect the direction of a parabola?
5. How would you find the range of a more complex function?

**Tip:** When dealing with quadratic functions, always identify the vertex, as it gives critical information about the range, especially when the parabola opens upwards or downwards.

Discover how to find the range of the quadratic function g(x) = -(x+6)^2 + 1. This solution explains how to identify the vertex and analyze the downward-opening parabola to determine the maximum value. Ideal for students in Grades 9-11 studying Algebra and Quadratic Functions.

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