Find the Domain of (x-2)²+1: Analyzing Positive and Negative Regions

Find the positive and negative domains of the function below:

$y=\left(x-2\right)^2+1$

To solve this problem, we'll analyze the function $y = (x-2)^2 + 1$ step-by-step:

• Step 1: Identify the function in its vertex form. The function is already presented as $(x-2)^2 + 1$, with vertex at $(2, 1)$.

• Step 2: Analyze the minimum point. The vertex represents the minimum value of the quadratic because the coefficient of the squared term, 1, is positive, meaning the parabola opens upwards.

• Step 3: Calculate the minimum value. By substituting $x = 2$ into the function, we find $y = (2-2)^2 + 1 = 1$.

• Step 4: Determine the range. Since the minimum value is 1, which is positive, the function never takes negative values. The range of $y$ is $(1, \infty)$.

• Step 5: Establish positive and negative domains: - Negative domain: The function does not have any negative values since it is always at or above 1. - Positive domain: The function is positive for all $x$, because the minimum value itself (1) is positive.

Therefore:

$x < 0 :$ none

$x > 0 :$ all $x$