Find the Positive Domain: Analyzing y=(x-2)² Function

Question

What is the positive domain of the function below?

$y=(x-2)^2$

00:00 Find the positive domain of the function

00:03 Positive domain means above the X-axis

00:06 For this, we need to find the intersection points with the X-axis

00:09 At intersection points with X-axis, Y=0, we'll substitute and solve

00:13 Take the square root to eliminate the exponent

00:17 Isolate X

00:24 This is the intersection point with the X-axis

00:32 Use the shortened multiplication formulas and expand the brackets

00:36 Notice the coefficient of X squared is positive

00:39 When the coefficient is positive, the function is smiling

00:45 Draw the function based on intersection points and function type

00:55 The function is positive while it's above the X-axis

01:06 And this is the solution to the question

In the first step, we place 0 in place of Y:

0 = (x-2)²

We perform a square root:

0=x-2

x=2

And thus we reveal the point

(2, 0)

This is the vertex of the parabola.

Then we decompose the equation into standard form:

y=(x-2)²

y=x²-4x+2

Since the coefficient of x² is positive, we learn that the parabola is a minimum parabola (smiling).

If we plot the parabola, it seems that it is actually positive except for its vertex.

Therefore the domain of positivity is all X, except X≠2.

all x, $x\ne2$