If M(t)=29.9t^2+3892, where t=0 represents 1990. How would I describe the transformation of the common function f(t)=t^2.

How can I rewrite the function so that t=0 represents 2000.

To describe the transformation of the common function f(t) = t^2 to the function M(t) = 29.9t^2 + 3892, we can break it down into two parts: horizontal and vertical transformations.

1. Horizontal transformation:
In the original function f(t) = t^2, when t=0 represents 1990, we can say that the original function is shifted 10 years to the right. To achieve this, we need to subtract 10 from the value of t. Therefore, the horizontal transformation is given by t' = t - 10.

2. Vertical transformation:
In the original function f(t) = t^2, there is no vertical transformation involved as there is no addition or subtraction outside the function. Thus, the vertical transformation remains the same.

To rewrite the function M(t) = 29.9t^2 + 3892 so that t=0 represents 2000, we need to apply the same horizontal transformation as before. Since the original function f(t) has been shifted 10 years to the right, we can similarly shift M(t) 10 years to the right. This can be accomplished by replacing t in the equation with t' = t - 10.

Therefore, the rewritten function will be:
M(t') = 29.9(t' + 10)^2 + 3892

Now, when t'=0, it represents the year 2000.