What is the range of this quadratic function?

y

=

–

x

2

–

12

x

–

32

{

y

|

y

≤

–

6

}

{

y

|

y

≤

4

}

{

y

|

y

≥

4

}


To determine the range of the quadratic function \( y = -x^2 - 12x - 32 \), we can follow these steps:

### Step 1: Determine the orientation of the parabola
Since the coefficient of \( x^2 \) is negative (\( -1 \)), the parabola opens downwards. This means that the function will have a maximum value.

### Step 2: Find the vertex
The vertex of the parabola will give us the maximum \( y \)-value since the parabola opens downwards. The \( x \)-coordinate of the vertex can be found using the formula:

\[
x = -\frac{b}{2a}
\]

Here, \( a = -1 \) and \( b = -12 \). Substituting these values:

\[
x = -\frac{-12}{2(-1)} = \frac{12}{-2} = -6
\]

Now, substitute this \( x \)-value back into the original function to find the corresponding \( y \)-value:

\[
y = -(-6)^2 - 12(-6) - 32
\]

\[
y = -36 + 72 - 32
\]

\[
y = 4
\]

So, the vertex of the parabola is at \( (-6, 4) \).

### Step 3: Determine the range
Since the parabola opens downwards and the maximum value of \( y \) is \( 4 \), the range of the function is all \( y \)-values less than or equal to 4.

### Final Answer:
The range of the quadratic function \( y = -x^2 - 12x - 32 \) is \( \{ y \mid y \leq 4 \} \).

Would you like more details or have any questions?

Here are 5 related questions:
1. How does the sign of the leading coefficient affect the range of a quadratic function?
2. How do you find the x-intercepts of a quadratic function?
3. What role does the vertex play in determining the range of a quadratic function?
4. How can you determine the axis of symmetry from the quadratic equation?
5. What does it mean for a parabola to have a maximum or minimum point?

**Tip:** Always check the sign of the leading coefficient first to understand whether the quadratic function has a maximum or minimum value.

Discover how to find the range of the quadratic function y = -x^2 - 12x - 32 with a step-by-step solution. This problem covers key concepts including quadratic functions, vertex form, and function ranges. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation