Calculate Area: Finding Ascending Regions of f(x) = -4x² - 24

Q: Why is the function increasing when x < 0?

When $ x , multiplying by -8 gives a positive result. Since \( f'(x) = -8x $, negative x-values make the derivative positive, meaning the function is going upward!

Q: How do I remember which direction the parabola opens?

Look at the coefficient of $ x^2 $! Since we have $ -4x^2 $, the negative coefficient means the parabola opens downward like an upside-down U.

Q: What's the difference between ascending area and maximum point?

Ascending area is the interval where the function is increasing (going up). The maximum point is the highest point on the graph. For $ f(x) = -4x^2 - 24 $, it increases on $ x and reaches maximum at \( x = 0 $.

Q: Can a function be always decreasing?

Yes! Some functions never increase. But this one isn't - $ f(x) = -4x^2 - 24 $ increases when $ x and decreases when \( x > 0 $.

Q: Why do I need to find the derivative?

The derivative $ f'(x) $ tells you the slope of the function at any point. When the slope is positive, the function is going upward (increasing)!

Quadratic Functions with Derivative Analysis

Find the ascending area of the function

$f(x)=-4x^2-24$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:06 Let's find where the function is increasing.

00:09 Notice that the coefficient of X squared is negative. This means the function curves downward.

00:21 First, let's look at the coefficients of the trinomial.

00:25 We'll use a formula to find the vertex of this parabola.

00:30 Substitute the given values into the formula. Solve for the vertex point.

00:35 This gives us the X value at the parabola's vertex.

00:39 Determine when the parabola goes up and down from this point.

00:44 Draw the X-axis to clearly find where the function increases.

00:56 And that's the solution to the question.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Find the ascending area of the function

$f(x)=-4x^2-24$

Step-by-step solution

To solve this problem, the focus is on determining the increasing intervals of the function $f(x) = -4x^2 - 24$ .

Here's how we'll proceed:

Step 1: Differentiate the function.
Step 2: Set the derivative greater than 0 to determine the intervals where the function ascends.
Step 3: Analyze the results to determine the intervals of increase.

Step 1: Find the derivative of $f(x) = -4x^2 - 24$ .
The derivative is a straightforward calculation:
$f'(x) = \frac{d}{dx}(-4x^2 - 24) = -8x$ .

Step 2: Solve $f'(x) > 0$ to find the increasing interval.
$-8x > 0$ leads to $x < 0$ .

Step 3: Conclude by analyzing this result.
This tells us that the function is increasing when $x < 0$ , meaning the ascending area of $f(x)$ lies in this interval.

Therefore, the solution to this problem is $x < 0$ .

Final Answer

$x < 0$

Key Points to Remember

Essential concepts to master this topic

Rule: Function increases where derivative is positive
Technique: Find f'(x) = -8x, then solve -8x > 0
Check: Test values: f'(-1) = 8 > 0 confirms increasing ✓

Common Mistakes

Avoid these frequent errors

Confusing increasing with maximum points
Don't look for where f'(x) = 0 to find increasing regions = this gives critical points, not intervals! Setting derivative equal to zero finds peaks and valleys. Always solve f'(x) > 0 for increasing intervals.

Practice Quiz

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One function

$$ y=-6x^2 $$

to the corresponding graph:

FAQ

Everything you need to know about this question

Why is the function increasing when x < 0?

When $x < 0$ , multiplying by -8 gives a positive result. Since $f'(x) = -8x$ , negative x-values make the derivative positive, meaning the function is going upward!

How do I remember which direction the parabola opens?

Look at the coefficient of $x^2$ ! Since we have $-4x^2$ , the negative coefficient means the parabola opens downward like an upside-down U.

What's the difference between ascending area and maximum point?

Ascending area is the interval where the function is increasing (going up). The maximum point is the highest point on the graph. For $f(x) = -4x^2 - 24$ , it increases on $x < 0$ and reaches maximum at $x = 0$ .

Can a function be always decreasing?

Yes! Some functions never increase. But this one isn't - $f(x) = -4x^2 - 24$ increases when $x < 0$ and decreases when $x > 0$ .

Why do I need to find the derivative?

The derivative $f'(x)$ tells you the slope of the function at any point. When the slope is positive, the function is going upward (increasing)!

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