Find Negative Area of y+4=(x+6)²: Quadratic Function Analysis

Video Solution

Solution Steps

00:00 Find the negative domain of the function

00:03 Use the abbreviated multiplication formulas and expand the brackets

00:09 Arrange the equation so it describes a function

00:15 Note that the coefficient of X squared is positive

00:24 When the coefficient is positive, the function smiles (opens upward)

00:30 Now we want to find the intersection points with the X-axis

00:36 At the intersection points with X-axis, Y=0, substitute and solve

00:41 Extract the root

00:47 When extracting a root, there are always 2 solutions (positive and negative)

00:50 Solve each possibility to find the points, isolate X

01:11 Draw the function according to intersection points and function type

01:29 The function is negative as long as it's below the X-axis

01:42 And this is the solution to the question

Step-by-Step Solution

To find the negative area of the function $y + 4 = (x + 6)^2$ , we need to determine where the function is below the x-axis, i.e., where $y < 0$ .

The equation can be rewritten as:

$y = (x + 6)^2 - 4$ .

We need to solve the inequality:

$(x + 6)^2 - 4 < 0$ .

Adding $4$ to both sides gives:

$(x + 6)^2 < 4$ .

Taking the square root gives:

$-2 < x + 6 < 2$ .

Subtracting $6$ from all sides results in:

$-8 < x < -4$ .

Thus, the interval where the function is below the x-axis is $-8 < x < -4$ .

This corresponds to answer choice 3 from the given options.