To find the maximum value of the function \( s(x) = -2x^2 - 4x \), we need to follow these steps:

### 1. Identify the form of the function:
This is a quadratic function of the form \( s(x) = ax^2 + bx + c \), where:
- \( a = -2 \) (the coefficient of \( x^2 \)),
- \( b = -4 \) (the coefficient of \( x \)),
- \( c = 0 \) (there is no constant term in this function).

Since the coefficient of \( x^2 \) is negative, the parabola opens downward, and thus the function has a maximum value at its vertex.

### 2. Find the vertex:
The x-coordinate of the vertex for a quadratic function \( ax^2 + bx + c \) can be found using the formula:

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

Substituting the values of \( a \) and \( b \):

\[
x = -\frac{-4}{2(-2)} = \frac{4}{-4} = -1
\]

### 3. Calculate the maximum value:
Now that we know the x-coordinate of the vertex is \( x = -1 \), we substitute this value back into the function \( s(x) \) to find the corresponding y-value (maximum value):

\[
s(-1) = -2(-1)^2 - 4(-1) = -2(1) + 4 = -2 + 4 = 2
\]

### Conclusion:
The maximum value of \( s(x) = -2x^2 - 4x \) is **2**, and it occurs at \( x = -1 \).

Would you like further details on any part of this? Here are 5 related questions to expand your understanding:

1. How do you find the maximum or minimum of a quadratic function?
2. What does the vertex of a parabola represent?
3. How do you determine whether a quadratic function opens upwards or downwards?
4. Can a quadratic function have both a maximum and a minimum value?
5. What are real-world examples of quadratic functions?

**Tip:** The sign of the leading coefficient \( a \) in a quadratic function determines whether the parabola opens up (positive) or down (negative), which in turn determines whether the function has a minimum or a maximum value.

Learn how to find the maximum value of the quadratic function s(x) = -2x^2 - 4x by using the vertex formula. This problem covers key algebraic concepts like quadratic functions and how to determine the vertex for maximum or minimum values. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation