Uncover Function Domains: Analyzing y=-2x²-8x-8

Q: What's the difference between domain and positive/negative domains?

The domain is all x-values where the function is defined. Positive/negative domains are subsets where the function output is positive or negative. For $ y = -2x^2 - 8x - 8 $, the domain is all real numbers, but it's only negative (except at x = -2).

Q: Why does the parabola opening downward matter?

Since a = -2 , the parabola opens downward. This means the function is negative everywhere except at its vertex. If it opened upward (a > 0), it would be positive except between its roots.

Q: What does it mean to have a double root at x = -2?

A double root means the parabola just touches the x-axis at that point but doesn't cross it. The function equals zero only at $ x = -2 $ and is negative everywhere else.

Q: Why do we exclude x = -2 from the negative domain?

At $ x = -2 $, the function equals zero, not negative. We say \( x because we're looking for where the function is strictly negative, and zero is neither positive nor negative.

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Find the positive and negative domains of the function below:

$y=-2x^2-8x-8$

2

Step-by-step solution

To find the positive and negative domains of the quadratic function $y = -2x^2 - 8x - 8$ , we first find the roots of the equation by using the quadratic formula.

The quadratic formula is given by:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

From the equation $-2x^2 - 8x - 8 = 0$ , we identify:

$a = -2$
$b = -8$
$c = -8$

Now substitute these into the quadratic formula:

x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(-2)(-8)}}{2(-2)}

This simplifies to:

x = \frac{8 \pm \sqrt{64 - 64}}{-4}

Which further simplifies to:

x = \frac{8 \pm \sqrt{0}}{-4}

Thus,

x = \frac{8}{-4}

The roots are:

x = -2

Since this is a parabola opening downwards (because $a < 0$ ), it is positive between its roots only if there are two distinct roots, and negative outside these roots. Here, with a double root, the function touches the x-axis and does not cross it. Thus, there are no positive intervals for $x > 0$ . The function is negative for all $x$ outside of $x = -2$ .

Therefore, the positive and negative domains are:

$x < 0 : x \ne -2$

$x > 0 :$ none

This matches choice 3 from the list of options.

3

Final Answer

$x < 0 : x\ne-2$

$x > 0 :$ none

Key Points to Remember

Essential concepts to master this topic

Rule: Find where function equals zero using quadratic formula
Technique: Check sign of leading coefficient: a = -2 < 0 means parabola opens downward
Check: Substitute test values into original function to verify positive/negative regions ✓

Common Mistakes

Avoid these frequent errors

Confusing function sign with domain restrictions
Don't think negative function values mean the function is undefined = wrong domain analysis! A function can have negative outputs and still be defined everywhere. Always remember that domain refers to valid x-values, while positive/negative domains refer to where the function output is positive or negative.

Practice Quiz

Test your knowledge with interactive questions

The graph of the function below intersects the X-axis at points A and B.

The vertex of the parabola is marked at point C.

Find all values of $$ x $$ where $f\left(x\right) > 0$ .

FAQ

Everything you need to know about this question

What's the difference between domain and positive/negative domains?

+

The domain is all x-values where the function is defined. Positive/negative domains are subsets where the function output is positive or negative. For $y = -2x^2 - 8x - 8$ , the domain is all real numbers, but it's only negative (except at x = -2).

Why does the parabola opening downward matter?

+

Since a = -2 < 0, the parabola opens downward. This means the function is negative everywhere except at its vertex. If it opened upward (a > 0), it would be positive except between its roots.

What does it mean to have a double root at x = -2?

+

A double root means the parabola just touches the x-axis at that point but doesn't cross it. The function equals zero only at $x = -2$ and is negative everywhere else.

How do I know there are no positive regions for x > 0?

+

Since the parabola opens downward and only touches the x-axis at $x = -2$ , it's below the x-axis everywhere else. Test any positive x-value: when $x = 1$ , $y = -2(1)^2 - 8(1) - 8 = -18 < 0$ .

Why do we exclude x = -2 from the negative domain?

+

At $x = -2$ , the function equals zero, not negative. We say $x < 0: x \ne -2$ because we're looking for where the function is strictly negative, and zero is neither positive nor negative.

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Uncover Function Domains: Analyzing y=-2x²-8x-8

Quadratic Functions with Sign Analysis

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Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

What's the difference between domain and positive/negative domains?

Why does the parabola opening downward matter?

What does it mean to have a double root at x = -2?

How do I know there are no positive regions for x > 0?

Why do we exclude x = -2 from the negative domain?

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