Calculate Negative Area: Finding Area Under y=-(x+4)²

Video Solution

Solution Steps

00:00 Find the domain of negativity in the function

00:03 We'll use the shortened multiplication formulas and open the parentheses

00:09 Negative times positive is always negative

00:19 We'll look at the coefficient of X squared, negative

00:23 When the coefficient is negative, the function is concave down

00:28 Now we want to find the intersection points with the X-axis

00:32 At the intersection point with X-axis, Y=0, we'll substitute and solve

00:39 We'll change from negative to positive

00:43 We'll take the square root to eliminate the exponent

00:48 Isolate X

00:51 This is the X value at the intersection point with X-axis

01:01 We'll draw the function according to the intersection points and function type

01:06 The function is negative while it's below the X-axis

01:11 And this is the solution to the problem

Step-by-Step Solution

To find the negative area for the function $y = -(x+4)^2$ , consider when the function is negative or equals zero:

The parabola $y = -(x+4)^2$ opens downward with the vertex at $x = -4$ , when $x = -4$ , $y = 0$ .
For $x \neq -4$ , the expression $-(x+4)^2$ is negative because $(x + 4)^2$ is always positive or zero, and the negative sign flips it to non-positive, i.e., negative unless zero.
Hence, everywhere except at the vertex $x = -4$ , the function is negative.

The function $y = -(x+4)^2$ is negative for every point except where $x = -4$ .

Therefore, the correct answer from the choice given is: For each $x \neq -4$ .