Calculate Negative Area of f(x)=-x²: Quadratic Function Analysis

Q: Why is the function negative for both positive and negative x values?

When you square any nonzero number (positive or negative), you get a positive result. Then multiplying by -1 makes it negative. So $ (-3)^2 = 9 $, then $ -9 $ is negative, and $ (2)^2 = 4 $, then $ -4 $ is also negative.

Q: What happens exactly at x = 0?

At $ x = 0 $, we get $ f(0) = -(0)^2 = 0 $. This is the vertex of the parabola where the function value is exactly zero, not positive or negative.

Q: How do I visualize this downward parabola?

Picture an upside-down U shape with the highest point at (0,0). The parabola opens downward, so everywhere except the very top point has negative y-values.

Q: Is this the same as finding where the graph is below the x-axis?

Exactly! The negative area refers to where the graph lies below the x-axis. For $ f(x) = -x^2 $, this happens everywhere except at the vertex.

Q: Why isn't the answer just 'all real numbers'?

Because at $ x = 0 $, the function value is zero, not negative. We need strictly negative values, so we exclude the point where $ x = 0 $.

Quadratic Functions with Negative Value Analysis

Find the negative area of the function

$f(x)=-x^2$

❤️ Continue Your Math Journey!

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

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the negative domain of the function

00:03 Notice the coefficient of X squared is negative, so the function is concave down

00:08 The negative domain is actually below the X-axis

00:12 Therefore, substitute Y=0 to find the intersection points with the X-axis

00:18 This is the intersection point of the function with the X-axis

00:25 Let's mark the intersection points with the X-axis

00:30 The negative domain is below the X-axis

00:39 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Find the negative area of the function

$f(x)=-x^2$

Step-by-step solution

To solve this problem, follow these steps:

Step 1: Understand that the function $f(x) = -x^2$ describes a downward-facing parabola with its vertex at the origin (0,0).
Step 2: Analyze where the function has negative values. For $f(x) = -x^2$ , any value of $x \neq 0$ will yield negative output since squaring any nonzero value is positive and multiplying by $-1$ gives a negative result.
Step 3: Recognize that the point $x = 0$ results in $f(0) = 0$ , meaning exactly at the vertex there is no negative value.

From these observations, you can conclude that $f(x)$ is negative for all $x \neq 0$ .

Therefore, the solution to the problem is $x \neq 0$ .

Final Answer

x≠0

Key Points to Remember

Essential concepts to master this topic

Rule: For $f(x) = -x^2$ , function is negative when $x \neq 0$
Technique: Test values: $f(2) = -4$ and $f(-3) = -9$ are both negative
Check: Only at vertex $x = 0$ , function equals zero: $f(0) = 0$ ✓

Common Mistakes

Avoid these frequent errors

Thinking negative area means x < 0
Don't assume negative area only occurs for negative x values = missing half the solution! The function $f(x) = -x^2$ gives negative outputs for both positive AND negative x values. Always remember that squaring eliminates the sign, so both x = 2 and x = -2 give negative function values.

Practice Quiz

Test your knowledge with interactive questions

Which chart represents the function $$ y=x^2-9 $$ ?

FAQ

Everything you need to know about this question

Why is the function negative for both positive and negative x values?

When you square any nonzero number (positive or negative), you get a positive result. Then multiplying by -1 makes it negative. So $(-3)^2 = 9$ , then $-9$ is negative, and $(2)^2 = 4$ , then $-4$ is also negative.

What happens exactly at x = 0?

At $x = 0$ , we get $f(0) = -(0)^2 = 0$ . This is the vertex of the parabola where the function value is exactly zero, not positive or negative.

How do I visualize this downward parabola?

Picture an upside-down U shape with the highest point at (0,0). The parabola opens downward, so everywhere except the very top point has negative y-values.

Is this the same as finding where the graph is below the x-axis?

Exactly! The negative area refers to where the graph lies below the x-axis. For $f(x) = -x^2$ , this happens everywhere except at the vertex.

Why isn't the answer just 'all real numbers'?

Because at $x = 0$ , the function value is zero, not negative. We need strictly negative values, so we exclude the point where $x = 0$ .

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Parabola Families 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