Calculate Area Under (x+5)²: Finding the Descending Region

Q: Why do we need the derivative to find where a function is decreasing?

The derivative tells us the slope at any point! When f'(x) downward (decreasing).

Q: How do I remember which inequality sign to use?

Think of it this way: Decreasing = Down = Negative slope. So use f'(x) 0 for increasing intervals.

Q: What does the vertex of this parabola have to do with the answer?

The vertex is at x = -5 where f'(x) = 0. Since this is a parabola opening upward, it decreases before the vertex (x -5).

Q: Can I solve this without using derivatives?

Yes! Since $ y = (x+5)^2 $ is a parabola with vertex at (-5, 0), you know it decreases to the left of the vertex and increases to the right.

Q: Why is the answer x -5?

The parabola opens upward (positive coefficient), so it goes down as you move left from the vertex at x = -5, then up as you move right. Left of -5 means x

Quadratic Functions with Decreasing Intervals

Find the descending area of the function

$y=(x+5)^2$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:06 Let's find the domain where the function decreases.

00:10 First, use multiplication formulas to expand the brackets.

00:14 Notice the coefficient of X squared. It's positive, so it's a happy or smiling function.

00:21 Next, examine each of the function's coefficients.

00:26 Use the formula to find the vertex point of the parabola.

00:37 Substitute the right values into the equation and solve for the vertex point.

00:46 This X value is the vertex point of the parabola.

00:50 In a U-shaped graph, the domain decreases before this vertex.

00:56 And that’s how we solve this problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Find the descending area of the function

$y=(x+5)^2$

Step-by-step solution

To solve this problem, we will determine when the function $y = (x+5)^2$ is decreasing by using its derivative:

Step 1: Differentiate $y = (x+5)^2$ with respect to $x$ , yielding $\frac{dy}{dx} = 2(x+5)$ .
Step 2: Set the derivative less than zero: $2(x+5) < 0$ .
Step 3: Simplify the inequality to solve for $x$ :
$x + 5 < 0$ implies $x < -5$ .

The function is decreasing for values of $x$ that satisfy $x < -5$ .

Therefore, the solution is $x < -5$ .

Final Answer

$x < -5$

Key Points to Remember

Essential concepts to master this topic

Derivative Rule: For decreasing intervals, find where f'(x) < 0
Technique: From y = (x+5)², get f'(x) = 2(x+5) < 0
Check: Test x = -6: f'(-6) = 2(-1) = -2 < 0 ✓

Common Mistakes

Avoid these frequent errors

Confusing decreasing with increasing intervals
Don't set the derivative greater than zero f'(x) > 0 for decreasing regions = gives increasing intervals instead! This gives the opposite answer. Always use f'(x) < 0 for decreasing and f'(x) > 0 for increasing.

Practice Quiz

Test your knowledge with interactive questions

Find the intersection of the function

$$ y=(x+4)^2 $$

With the Y

FAQ

Everything you need to know about this question

Why do we need the derivative to find where a function is decreasing?

The derivative tells us the slope at any point! When f'(x) < 0, the slope is negative, meaning the function is going downward (decreasing).

How do I remember which inequality sign to use?

Think of it this way: Decreasing = Down = Negative slope. So use f'(x) < 0 for decreasing intervals and f'(x) > 0 for increasing intervals.

What does the vertex of this parabola have to do with the answer?

The vertex is at x = -5 where f'(x) = 0. Since this is a parabola opening upward, it decreases before the vertex (x < -5) and increases after (x > -5).

Can I solve this without using derivatives?

Yes! Since $y = (x+5)^2$ is a parabola with vertex at (-5, 0), you know it decreases to the left of the vertex and increases to the right.

Why is the answer x < -5 and not x > -5?

The parabola opens upward (positive coefficient), so it goes down as you move left from the vertex at x = -5, then up as you move right. Left of -5 means x < -5.

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