Calculate Area Under y=-(x+3)²+25: Quadratic Function Analysis

Q: How do I know the parabola is positive between the roots?

Since this parabola opens downward (coefficient of $ x^2 $ is negative), it's like an upside-down U. The vertex is at the top, so values between the roots are positive.

Q: Why is this called 'positive area' instead of just 'positive values'?

The term positive area refers to the region where the function is above the x-axis (y > 0). This creates a physical 'area' between the curve and the x-axis that we can measure.

Quadratic Inequalities with Interval Notation

Find the positive area of the function

$y=-(x+3)^2+25$

❤️ 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

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Find the positive area of the function

$y=-(x+3)^2+25$

Step-by-step solution

To solve this problem, follow these steps:

Step 1: Find the x-intercepts by solving the equation $-(x+3)^2 + 25 = 0$ .
Step 2: Determine when the expression $-(x+3)^2 + 25$ is positive.
Step 3: Verify the correct answer choice based on the solution.

Now let's work through each step:

Step 1: Solve $-(x+3)^2 + 25 = 0$ .

Rewriting, we have:

$-(x+3)^2 = -25$

$(x+3)^2 = 25$

This gives $x+3 = \pm 5$ .

Solving these, we find $x = 2$ and $x = -8$ .

Step 2: Determine when $-(x+3)^2 + 25 > 0$ .

We know that the parabola is downward opening, so it will be positive between the roots found, $-8$ and $2$ .

Thus, the positive interval is $-8 < x < 2$ .

Step 3: Verify the correct answer choice.

The correct answer, matching the solution above, is choice 1: $-8 < x < 2$ .

Therefore, the positive area of the function occurs in the interval $-8 < x < 2$ .

Final Answer

$-8 < x < 2$

Key Points to Remember

Essential concepts to master this topic

Zero-Finding: Set quadratic equal to zero to find boundary points
Sign Analysis: Test values between roots: $-(0+3)^2+25 = 16 > 0$
Interval Check: Verify endpoints excluded since function equals zero there ✓

Common Mistakes

Avoid these frequent errors

Including boundary points in the interval
Don't write ≤ or ≥ when the function equals zero at x = -8 and x = 2! This includes points where area is zero, not positive. Always use strict inequalities < and > for positive area regions.

Practice Quiz

Test your knowledge with interactive questions

Find the corresponding algebraic representation of the drawing:

FAQ

Everything you need to know about this question

Why do we exclude the x-intercepts from the positive area?

At the x-intercepts (x = -8 and x = 2), the function value is zero, not positive! We want the interval where $y > 0$ , so we use strict inequality signs.

How do I know the parabola is positive between the roots?

Since this parabola opens downward (coefficient of $x^2$ is negative), it's like an upside-down U. The vertex is at the top, so values between the roots are positive.

What if I can't remember the vertex form?

You can always test a point! Pick any x-value between -8 and 2 (like x = 0) and substitute: $-(0+3)^2+25 = 16 > 0$ . Positive result confirms the interval is correct.

Why is this called 'positive area' instead of just 'positive values'?

The term positive area refers to the region where the function is above the x-axis (y > 0). This creates a physical 'area' between the curve and the x-axis that we can measure.

Could the answer ever include infinity symbols?

Not for this type of downward parabola! Since it has two real roots, the positive region is always a bounded interval between the roots, never extending to infinity.

🌟 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