4-1:Exponential 4-1:Exponential Growth and DecayGrowth and Decay4-1:Exponential 4-1:Exponential

Growth and DecayGrowth and DecayEnglish CasbarroEnglish Casbarro

Unit 4Unit 4

Exponential Functions

If an output doubles every year, it can be modeled by an exponential function

The parent function is: f(x) = bx, where b is a constant and x is the independent variable.

b > 0, and b ≠ 1

Remember the graph on the introduction: y = 2x

Notice that as x decreases, the graph gets closer and closer to the x-axis.The graph, however, will never touch the x-axis, so it is an asymptote.

The standard form of an exponential growth (decay) function:

a is the initial amount b is the constant of growth if b > 1b is the constant of decay if 0< b < 1 x is usually the time

For exponential growth, as the value of x increases, the value of y increases. For exponential decay, as the value of x increases, the value of y decreases, approaching zero.

You can model growth or decay by a constant percent increase or decrease withthe following formula:

Final amount

Initial amount

Rate of growth or decay

Number of time periods

In the formula, the base of the exponential expression 1 + r, is called the growth factor. Similarly, 1 – r is called the decay factor.

The value of a truck bought new for $28,000 decreases 9.5% each year.Write an exponential function, and graph the function. Use the graph toPredict when the value will fall to $5000.

Transformation of the graph

Turn in the following problems1. The compound interest formula is , where A is the amount earned, P is the principal, r is the annual interest rate, t is the time in years, and n is the number of compounding periods per year. Harry invested $5000 at 5% interest compounded quarterly(4 times per year). a. How much will the investment be worth after 5 years? b. When will the investment be worth more than $10,000? c. What if Harry could have invested the same amount in an account that paid 5% interest compounded monthly (12 times per year). How much more would his investment have been worth after 5 years?

2. What are the values of a and b in f(x) = abx in the graph shown at the right?

3. The population of Midland, Texas was 89, 443 in 1990 and has increased at a rate of 0.6% per year since then. Which function represents the Midland’s growth function after t years?

A. 89,443(1.6)t B. 89,443(1.06)t

C. 89,443(1.006)t D. 89,443(1.0006)t