Calculate Negative Area: Finding Area Below y=(x-1)²-4

Find the negative area of the function

$y=(x-1)^2-4$

To solve this problem, we'll consider the steps as follows:

• Step 1: Recognize the given function as a standard parabola centered at $(h, k) = (1, -4)$ that opens upwards.
• Step 2: The formula $y = (x-1)^2 - 4$ represents the parabola with vertex at $(1, -4)$.
• Step 3: Determine roots by solving the equation $(x-1)^2 - 4 = 0$.
• Step 4: Rearrange to $(x-1)^2 = 4$ leading to $x - 1 = \pm 2$.
• Step 5: Solve these to find roots: $x = 3$ and $x = -1$.
• Step 6: Calculate regions where the function is negative by testing intervals between the roots.

Now, we apply these steps:

Step 1: The function, expressed in its vertex form, indicates a parabola opening upwards (since the coefficient of $(x-1)^2$ is positive).

Step 2: The equation for determining where the parabola touches the x-axis is $(x−1)^2−4=0$. Solving this, we rearrange it to $(x-1)^2 = 4$.

Step 3: Solving for $(x-1)^2 = 4$, gives $x - 1 = 2$ or $x - 1 = -2$, leading to solutions $x = 3$ and $x = -1$.

Step 4: The parabola is negative between these roots. So, the inequality $(x-1)^2 < 4$ holds true for $-1 < x < 3$.

Therefore, the negative area of the function lies in the interval $-1 < x < 3$.

Conclusively, the negative domain of the function is $-1 < x < 3$.