Calculate Area Under f(x) = -x² + 9: Quadratic Function Analysis

Q: What does 'positive area' actually mean for this function?

Positive area means the region where the function is above the x-axis, so f(x) > 0. For $ f(x) = -x^2 + 9 $, this creates an upside-down parabola that's positive between its roots.

Q: Why do we set the function equal to zero first?

Setting $ -x^2 + 9 = 0 $ finds the x-intercepts where the parabola crosses the x-axis. These points divide the number line into intervals where the function is either positive or negative.

Q: How do I know which interval to test?

Pick any convenient number from each interval created by the roots. For example, between -3 and 3, choose x = 0 because it makes calculations easy: $ f(0) = 9 $.

Q: Could this parabola ever be positive outside the interval (-3, 3)?

No! Since the coefficient of $ x^2 $ is negative (-1), this parabola opens downward. It can only be positive between its two roots, never outside them.

Q: What if I got the wrong sign when testing intervals?

Double-check your arithmetic when substituting test valuesRemember: $ (-4)^2 = 16 $, so $ f(-4) = -16 + 9 = -7 $The pattern should be: negative, positive, negative for this downward parabola

Quadratic Functions with Sign Analysis

Find the positive area of the function
$f(x)=-x^2+9$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:05 Let's find where the function is positive.

00:09 The X squared term has a negative coefficient, so the graph curves downward.

00:18 The positive domain is above the X-axis.

00:22 We'll set Y equal to zero to find where it crosses the X-axis.

00:29 Next, let's solve for X.

00:36 Take the square root of both sides.

00:40 Remember, there are two square root solutions: positive and negative.

00:46 These give us the points where it intersects the X-axis.

00:56 Now, let's put these intersection points on the graph.

01:10 Below the X-axis, the function is negative.

01:15 Above the X-axis, the function is positive.

01:19 So, let's find where the function stays positive.

01:23 And that's how we solve the problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Find the positive area of the function
$f(x)=-x^2+9$

Step-by-step solution

To determine the positive area of the function $f(x) = -x^2 + 9$ , we follow these steps:

Step 1: Find the roots of the equation $-x^2 + 9 = 0$ .
Step 2: Solve the equation for the roots:

$-x^2 + 9 = 0$

$x^2 = 9$

$x = \pm 3$

Step 3: Identify the intervals defined by these roots: $(-\infty, -3)$ , $(-3, 3)$ , and $(3, \infty)$ .
Step 4: Test the sign of $f(x)$ in each interval:

For interval $(-3, 3)$ : Choose $x = 0$ , $f(0) = 9$ (positive)

For interval $(-\infty, -3)$ : Choose $x = -4$ , $f(-4) = -7$ (negative)

For interval $(3, \infty)$ : Choose $x = 4$ , $f(4) = -7$ (negative)

Step 5: Conclude the positive area is for $-3 < x < 3$ .

The positive area of the function is described by $-3 < x < 3$ , which corresponds to the second choice.

Therefore, the solution to the problem is $-3 < x < 3$ .

Final Answer

$-3 < x < 3$

Key Points to Remember

Essential concepts to master this topic

Zero Rule: Set function equal to zero to find where it crosses x-axis
Technique: Test sign in each interval: f(0) = 9 is positive
Check: Verify endpoints give zero: f(-3) = 0 and f(3) = 0 ✓

Common Mistakes

Avoid these frequent errors

Confusing where function is positive with where area is positive
Don't just find where f(x) > 0 without understanding what 'positive area' means = missing the connection to intervals! Students often list intervals correctly but don't realize positive area means the region above the x-axis. Always connect the algebraic solution to the geometric meaning of area under the curve.

Practice Quiz

Test your knowledge with interactive questions

Find the ascending area of the function

$$ f(x)=2x^2 $$

FAQ

Everything you need to know about this question

What does 'positive area' actually mean for this function?

Positive area means the region where the function is above the x-axis, so f(x) > 0. For $f(x) = -x^2 + 9$ , this creates an upside-down parabola that's positive between its roots.

Why do we set the function equal to zero first?

Setting $-x^2 + 9 = 0$ finds the x-intercepts where the parabola crosses the x-axis. These points divide the number line into intervals where the function is either positive or negative.

How do I know which interval to test?

Pick any convenient number from each interval created by the roots. For example, between -3 and 3, choose x = 0 because it makes calculations easy: $f(0) = 9$ .

Could this parabola ever be positive outside the interval (-3, 3)?

No! Since the coefficient of $x^2$ is negative (-1), this parabola opens downward. It can only be positive between its two roots, never outside them.

What if I got the wrong sign when testing intervals?

Double-check your arithmetic when substituting test values
Remember: $(-4)^2 = 16$ , so $f(-4) = -16 + 9 = -7$
The pattern should be: negative, positive, negative for this downward parabola

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