Question

1. The amount M (in trillions of dollars) of mortgage debt outstanding in the United States from 1990through 2009 can be approximated by the function M = f() = 0.0037(t + 14.979), where t = 0represents the year 1990.(a) Describe the transformation of the common function f(x) = x2. Then sketch the graph over theinterval 0 Sis 19.(b) Rewrite the function so that t = 0 represents the year 2000. Explain how you got your answer.(c) Use the model from part (b) to predict the mortgage debt in the year 2014

Answer 1

a)

To determine which transformation from the function:

`f(x)=x^2`

we are making we need to remember that:

From the table we conclude that to get function M we need to do the following trasnformations:

• Translate function f 14.979 units to the left.

,

• Vertical compression with a factor of 0.0037

The sketch of the graph is shown below:

b)

To rewrite the function so that t=0 represents the year 2000 we need to translate the function M 10 units to the left, that is we add 10 to the variable t, then we have:

`\begin{gathered} M(t)=0.0037\mleft(t+10+14.979\mright)^2 \\ =0.0037(t+24.979)^2 \end{gathered}`

Therefore the function we need here is:

`M(t)=0.0037(t+24.979)^2`

c)

For the year 2014 fourteen years have passed from 2000; plugging 14 in the model in part b we have:

`\begin{gathered} M(14)=0.0037(14+24.979)^2 \\ =5.62 \end{gathered}`

Therefore in 2014 the debt is 5.62 trillions of dollars.