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

Question

Find the descending area of the function

$y=(x+5)^2$

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!

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 $x < -5$ .

$x < -5$