For the transformation t, write the t-1. t: (x, y) ( 2x, y + 5) t -1 (x, y) (x - 2, y - 5) (½x, y - 5) (2x - 1, y + 4)

Choice B) (1/2x, y - 5)

The original transformation T takes some point (x,y) and turns it into (2x, y+5). What's going on with the coordinates? Well x is doubled to 2x while y is having 5 added to it. To get the inverse, we simply undo everything happening to the coordinates. We do the opposite. 

Going from x to 2x has us multiply by 2. To backwards, we divide by 2. This is the same as multiplying by 1/2. At this point, the only answer is B. So we could stop here if we wanted. But let's keep going.

To go from y to y+5, we add on 5. Undo this by subtracting 5. So the inverse would be y-5. 

That's why the inverse transformation is (1/2x, y-5)

option A. t ≥ 6

Explanation:

The function D = 0.0625t² − 0.75t + 4.25  is a quadratic function; thus, its graph is a parabola.

Since the coefficient of the quadratic term is positive, the parabola opens upward and the vertex is the minimum point of the curve.

That means that the curve decreases, reachs the minimum and then increases.

Then, the demand will increase from the time equal to the x-coordinate (t) of the of the vertex onwards.

Determine the vertex:

The x-coordinate of the vertex of a parabola given by the general form y = ax² + bx + c is equal to -b/(2a).

For D = 0.0625t² − 0.75t + 4.25, b = -0.75 and a = 0.0625, then:

x-coordinate of the vertex = - (-0.75)/(2 × 0.0625) = 6.

Therefore, the demand will increase from the month 6 onwards. That is t ≥ 6 (option A.).