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

Quadratic Functions with Sign Analysis

Find the positive and negative domains of the function below:

y=−2x2−8x−8 y=-2x^2-8x-8

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

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=−2x2−8x−8 y=-2x^2-8x-8

2

Step-by-step solution

To find the positive and negative domains of the quadratic function y=−2x2−8x−8 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=−b±b2−4ac2a x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

From the equation −2x2−8x−8=0 -2x^2 - 8x - 8 = 0 , we identify:

  • a=−2 a = -2
  • b=−8 b = -8
  • c=−8 c = -8

Now substitute these into the quadratic formula:

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

This simplifies to:

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

Which further simplifies to:

x=8±0−4 x = \frac{8 \pm \sqrt{0}}{-4}

Thus,

x=8−4 x = \frac{8}{-4}

The roots are:

x=−2 x = -2

Since this is a parabola opening downwards (because a<0 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 x > 0 . The function is negative for all x x outside of x=−2 x = -2 .

Therefore, the positive and negative domains are:

x<0:x≠−2 x < 0 : x \ne -2

x>0: x > 0 : none

This matches choice 3 from the list of options.

3

Final Answer

x<0:x≠−2 x < 0 : x\ne-2

x>0: 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 does not intersect the \( x \)-axis.

The parabola's vertex is marked A.

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

AAAX

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=−2x2−8x−8 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 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 x = -2 , it's below the x-axis everywhere else. Test any positive x-value: when x=1 x = 1 , y=−2(1)2−8(1)−8=−18<0 y = -2(1)^2 - 8(1) - 8 = -18 < 0 .

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

+

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

🌟 Unlock Your Math Potential

Get unlimited access to all 18 The Quadratic Function questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime

More Questions

Click on any question to see the complete solution with step-by-step explanations

Positive and Negative Domains

Analyze the Domains of the Function: y = -1/3x² + 2/3x - 1/3
Click to solve
Determine Positive and Negative Domains of the Function: y = x² - 22x + 121
Click to solve

Standard Representation

Domain Analysis of y=x²-8x+16: Finding Positive and Negative Regions
Click to solve
Analyze x² + 10x + 25: Exploring Positive and Negative Domains
Click to solve