7 Pull It All Together

BIG idea Modeling

You can represent many real-world mathematical problems algebraically. An algebraic model can lead to an algebraic solution.

Task 1

Suppose you invest a dollars to earn an annual interest rate of r percent (as a decimal). After t years, the value of the investment with interest compounded yearly is A ( t ) = a ( 1 + r ) t . eh open t close equals eh . open 1 plus r close to the t . .  The value with interest compounded continuously is A ( t ) = a · e r t . eh open t close equals eh middle dot , e super r t end super , .

1. Explain why you can call e r − 1 e to the r , minus 1  the effective annual interest rate for the continuous compounding.
2. Suppose you can earn interest at some rate between 0% and 5%. Use your knowledge of the exponential function to explain why continuous compounding does not give you much of an investment advantage.
3. For each situation find the unknown quantity, such that continuous compounding gives you a $1 advantage over annually compounded interest.
   • How much must you invest for 1 year at 2%?
   • At what interest rate must you invest $1000 for 1 year?
   • For how long must you invest $1000 at 2%?

BIG idea Function

You can use transformations such as translations, reflections, and dilations to understand relationships within a family of functions.

Task 2

f ( x ) = b x f open x close equals , b to the x  and g ( x ) = log b x g open x close equals , log base b , x  are inverse functions. Explain why each of the following is true.

1. The translation f 1 ( x ) = b x − h f sub 1 , open x close equals . b super x minus h end super  of f is equivalent to a vertical stretch or compression of f.
2. The inverse of f 1 ( x ) = b x − h f sub 1 , open x close equals . b super x minus h end super  is equivalent to a translation of g.
3. The inverse of f 1 ( x ) = b x − h f sub 1 , open x close equals . b super x minus h end super  is not equivalent to a vertical stretch or compression of g.
4. The function h ( x ) = log c x h open x close equals , log base c , x  is a vertical stretch or compression of g or of its reflection − g . negative g .

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