Algebra 2 Exploring Exponential Models Lesson 7-1

Goals Goal To model exponential growth and decay. Rubric Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems.

Vocabulary Exponential Function Exponential Growth Exponential Decay Asymptote Growth Factor Decay Factor

Essential Question Big Idea: Modeling What is an exponential function?

Definition Exponential function - is a function of the form f(x) = ab x, where a ≠ 0, b is a positive number and b  1. The parent exponential function is f(x) = b x, where the base b is a constant and the exponent x is the independent variable.

Exponential Functions y = 2 x y = 3 x y = 2 x y = 1 - e x y = x 2 y = x 3 y = x 5 - 3x 2 + x y = 6 – 5x + x 2 Exponential Functions Power Functions Not Exponential

Negative exponents indicate a reciprocal. For example: Remember! In the function y = b x, y is a function of x because the value of y depends on the value of x. Remember!

The value of f(x) = 3 x when x = 2 is f(2) = 3 2 = The value of g(x) = 0.5 x when x = 4 is g(4) = = The value of f(x) = 3 x when x = –2 is 9 f(–2) = 3 –2 = Evaluating Exponential Functions

We’ve graphed linear and quadratic functions. We are now going to graph exponential functi ons. They are different than any of the other types of functions we’ve studied because the independent variable is in the exponent. Let’s look at the graph of this function by plotting some points. x 2 x /2 -2 1/4 -3 1/ Recall what a negative exponent means: BASE Exponential Graph

Example: Graph the exponential function using a table of values. xy /3 1/9

Your Turn: Graph the exponential function. xy /16 1/

EXAMPLE 2 124y 210– 1– 2x Your Turn: Graph the exponential function.

Notice as the x-values decrease, the graph of the function gets closer and closer to the x-axis. The function never reaches the x-axis because the value of 2 x cannot be zero. In this case, the x-axis is an asymptote. Asymptote - is a line that a graphed function approaches as the value of x gets very large or very small. Definition

A function of the form f(x) = ab x, with a > 0 and b > 1, is an exponential growth function, which increases as x increases. When 0 < b < 1, the function is called an exponential decay function, which decreases as x increases. Definition y x (0, 1) Domain: (– ,  ) Range: (0,  ) Horizontal Asymptote y = growth b > 1 y Range: (0,  ) x (0, 1) Domain: (– ,  ) Horizontal Asymptote y = decay 0 < b < 1

State “exponential growth” or “exponential decay” b > 0, exponential growth0 < b < 1, exponential decay b>1, exponential growth0 < b < 1, exponential decay

Tell whether the function shows growth or decay. Then graph. Step 1 Find the value of the base. The base,,is less than 1. This is an exponential decay function. Example:

Step 2 Graph the function by using a table of values. x f(x) Example: Continued

Tell whether the function shows growth or decay. Then graph. Step 1 Find the value of the base. g(x) = 100(1.05) x The base, 1.05, is greater than 1. This is an exponential growth function. g(x) = 100(1.05) x Your Turn:

Step 2 Graph the function by using a graphing calculator. Continued

Tell whether the function p(x) = 5(1.2 x ) shows growth or decay. Then graph. Step 1 Find the value of the base. p(x) = 5(1.2 x ) The base, 1.2, is greater than 1. This is an exponential growth function. Your Turn:

Step 2 Graph the function by using a table of values. x–12–8– f(x) Continued

Modeling Exponential Growth/Decay Growth Factor – the value of b for exponential growth y = ab x, with b > 1. –A quantity that exhibits exponential growth increases by a constant percentage each time period. The percent increase r, written as a decimal, is the rate of increase or growth rate. –For exponential growth b = 1 + r. Decay Factor – the value of b for exponential decay, with 0 < b < 1. –The quantity decreases by a constant percentage each time period. The percent decrease r, is the rate of decay. –For exponential decay b = 1 – r.

You can model growth or decay by a constant percent increase or decrease with the following formula: In the formula, the base of the exponential expression, 1 + r, is called the growth factor. Similarly, 1 – r is the decay factor. Modeling Exponential Growth/Decay

E XPONENTIAL G ROWTH M ODEL W RITING E XPONENTIAL G ROWTH M ODELS A quantity is growing exponentially if it increases by the same percent in each time period. a is the initial amount. t is the time period. (1 + r) is the growth factor, r is the growth rate. The percent of increase is 100r. f(x) = a (1 + r) t

E XPONENTIAL DECAY M ODEL W RITING E XPONENTIAL DECAY M ODELS A quantity is decaying exponentially if it decreases by the same percent in each time period. a is the initial amount. t is the time period. (1 - r) is the decay factor, r is the decay rate. The percent of decrease is 100r. f(x) = a (1 - r) t

Clara invests $5000 in an account that pays 6.25% interest per year. After how many years will her investment be worth $10,000? Step 1 Write a function to model the growth in value of her investment. f(t) = a(1 + r) t Exponential growth function. Substitute 5000 for a and for r. f(t) = 5000( ) t f(t) = 5000(1.0625) t Simplify. Example: Application

Step 2 When graphing exponential functions in an appropriate domain, you may need to adjust the range a few times to show the key points of the function. Step 3 Use the graph to predict when the value of the investment will reach $10,000. Use the feature to find the t-value where f(t) ≈ 10,000. Example: Continued The function value is approximately 10,000 when t ≈ The investment will be worth $10,000 about years after it was purchased.

A city population, which was initially 15,500, has been dropping 3% a year. Write an exponential function and graph the function. Use the graph to predict when the population will drop below f(t) = a(1 – r) t Substitute 15,500 for a and 0.03 for r. f(t) = 15,500(1 – 0.03) t f(t) = 15,500(0.97) t Simplify. Exponential decay function. Example:

Graph the function. Use to find when the population will fall below It will take about 22 years for the population to fall below , Example: Continued

In 1981, the Australian humpback whale population was 350 and increased at a rate of 14% each year since then. Write a function to model population growth. Use a graph to predict when the population will reach 20,000. P(t) = a(1 + r) t Substitute 350 for a and 0.14 for r. P(t) = 350( ) t P(t) = 350(1.14) t Simplify. Exponential growth function. Your Turn:

Graph the function. Use to find when the population will reach 20,000. It will take about 31 years for the population to reach 20,000. Continued

A motor scooter purchased for $1000 depreciates at an annual rate of 15%. Write an exponential function and graph the function. Use the graph to predict when the value will fall below $100. f(t) = a(1 – r) t Substitute 1,000 for a and 0.15 for r. f(t) = 1000(1 – 0.15) t f(t) = 1000(0.85) t Simplify. Exponential decay function. Your Turn:

Graph the function. Use to find when the value will fall below 100. It will take about 14.2 years for the value to fall below Continued

In 2000, the world population was 6.08 billion and was increasing at a rate 1.21% each year. 1. Write a function for world population. Does the function represent growth or decay? P(t) = 6.08(1.0121) t growth The value of a $3000 computer decreases about 30% each year. 2. Write a function for the computer’s value. Does the function represent growth or decay? V(t)≈ 3000(0.7) t decay More Practice

Writing an Exponential Function Procedure: 1.Define the variables. 2.Determine r. 3.Use r to determine b. 4.Write the exponential function f(x) = a(1 ± r) t. 5.Use the exponential function (model) to make a prediction.

EXAMPLE 5 The number of acres of Ponderosa pine forests decreased in the western United States from 1963 to 2002 by 0.5% annually. In 1963 there were about 41 million acres of Ponderosa pine forests. a. Write a function that models the number of acres of Ponderosa pine forests in the western United States over time. b. To the nearest tenth, about how many million acres of Ponderosa pine forests were there in 2002? Example:

Solution EXAMPLE 5 P = a(1 – r) t Write exponential decay model. = 41(1 –0.005) t Substitute 41 for a and for r. = 41(0.995) t Simplify. a.a. Let P be the number of acres (in millions), and let t be the time (in years) since The initial value is 41, and the decay rate is

P = 41(0.995) Solution Number of years from 1963 to 2002 (2002 – 1963 = 39) Substitute 39 for t. Use a calculator. ANSWER There were about 33.7 million acres of Ponderosa pine forests in b. To the nearest tenth, about how many million acres of Ponderosa pine forests were therein 2002 ?

WHAT IF? In the last Example, suppose the decay rate of the forests remains the same beyond About how many acres will be left in 2010? SOLUTION P = a(1 – r) t Write exponential decay model. = 41(1 –0.005) t Substitute 41 for a and for r. = 41(0.995) t Simplify. a.a. Let P be the number of acres (in millions), and let t be the time (in years) since The initial value is 41, and the decay rate is Example: Contin ued

GUIDED PRACTICE To find the number of acres will be left in 2010, 47 years after 1963, substitute 47 for t P = 41(0.995) 47 Substitute 47 for t. = 32.4 There will be about 32.4 million acres of Ponderosa pine forest in ANSWER Solution

A population of 20 rabbits is released into a wildlife region. The population triples each year for 5 years. Example:

A population of 20 rabbits is released into a wildlife region. The population triples each year for 5 years. a. What is the population after 5 years? Writing an Exponential Growth Model S OLUTION After 5 years, the population is P = a(1 + r) t Exponential growth model = 20(1 + 2) 5 = = 4860 Help Substitute a, r, and t. Simplify. Evaluate. There will be about 4860 rabbits after 5 years.

Writing an Exponential Decay Model C OMPOUND I NTEREST From 1982 through 1997, the purchasing power of a dollar decreased by about 3.5% per year. Using 1982 as the base for comparison, what was the purchasing power of a dollar in 1997? S OLUTION Let y represent the purchasing power and let t = 0 represent the year The initial amount is $1. Use an exponential decay model. = (1)(1 – 0.035) t = t y = a (1 – r) t y = Exponential decay model Substitute 1 for a, for r. Simplify. Because 1997 is 15 years after 1982, substitute 15 for t. Substitute 15 for t. The purchasing power of a dollar in 1997 compared to 1982 was $0.59.  0.59

Your business had a profit of $25,000 in If the profit increased by 12% each year, what would your expected profit be in the year 2010? Write down the equation you would use and solve. Your Turn:

Solution a = $25,000 t = 12 r = 0.12 Growth factor = 1.12

Iodine-131 is a radioactive isotope used in medicine. Its half-life or decay rate of 50% is 8 days. If a patient is given 25mg of iodine-131, how much would be left after 32 days or 4 half-lives. Write down the equation you would use and solve. Your Turn:

a = 25 mg t = 4 r = 0.5 Decay factor = 0.5 Solution

Essential Question Big Idea: Modeling What is an exponential function? An exponential function is a function in which the independent variable is the exponent. The general form is y = ab x, where a ≠ 0, b > 0, and b ≠ 1. The graph of an exponential function has a horizontal asymptote. Use an exponential function to model exponential growth or decay.

Assignment Section 7-1, Pg 467 – 469; #1 – 9 all, 10 – 36 even.