The function g(x) = -2x2 - 8x - 6 has a maximum or minimum value. What is the maximum or minimum value of g(x)?

Question 30Answer

a.

-6

b.

-8

c.

-10

d.

2

The function given is \( g(x) = -2x^2 - 8x - 6 \). 

This is a quadratic function of the form \( g(x) = ax^2 + bx + c \), where \( a = -2 \), \( b = -8 \), and \( c = -6 \). 

To determine whether the function has a maximum or minimum value, we need to examine the coefficient \( a \). Since \( a \) is negative (\( a = -2 \)), the parabola opens downwards, meaning the function has a **maximum** value.

The maximum or minimum value of a quadratic function occurs at the vertex of the parabola. The \( x \)-coordinate of the vertex can be found using the formula:

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

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

\[
x = \frac{-(-8)}{2(-2)} = \frac{8}{-4} = -2
\]

Now, substitute \( x = -2 \) back into the function to find the corresponding \( y \)-value:

\[
g(-2) = -2(-2)^2 - 8(-2) - 6 
\]

\[
g(-2) = -2(4) + 16 - 6 
\]

\[
g(-2) = -8 + 16 - 6 = 2
\]

So, the maximum value of \( g(x) \) is **2**.

The correct answer is:

**d. 2**

Would you like more details or have any questions about this problem? Here are some related questions:

1. How do you determine the vertex of a quadratic function?
2. What does the sign of the leading coefficient indicate about the graph of a quadratic function?
3. Can a quadratic function have both a maximum and minimum value?
4. How do you complete the square to find the vertex form of a quadratic function?
5. How can you find the axis of symmetry for a quadratic function?
6. What are the effects of changing the coefficient \( a \) in a quadratic function?
7. How do you determine if a quadratic equation has real roots?
8. What is the significance of the discriminant in quadratic equations?

**Tip:** For any quadratic function, the vertex formula \( x = \frac{-b}{2a} \) quickly gives you the \( x \)-value at which the function reaches its maximum or minimum.

Learn how to find the maximum or minimum value of the quadratic function g(x) = -2x^2 - 8x - 6 using the vertex formula and algebraic manipulation. Understand the concept of vertex for quadratic functions and how the sign of coefficient 'a' affects the parabola's shape. Suitable for high school students studying quadratic functions.

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