Holt Algebra Exponential Functions, Growth, and Decay exponential function baseasymptote exponential growth and decay Vocabulary Write and evaluate exponential expressions to model growth and decay situations. Objective

Holt Algebra Exponential Functions, Growth, and Decay Notes 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? 2. Predict the population in The value of a $3000 computer decreases about 30% each year. 3. Write a function for the computer’s value. Does the function represent growth or decay? 4. Predict the value in 4 years.

Holt Algebra Exponential Functions, Growth, and Decay Growth that doubles every year can be modeled by using a function with a variable as an exponent. This function is known as an exponential function. The parent exponential function is f(x) = b x, where the base b is a constant and the exponent x is the independent variable.

Holt Algebra Exponential Functions, Growth, and Decay The graph of the parent function f(x) = 2 x is shown. The domain is all real numbers and the range is {y|y > 0}.

Holt Algebra Exponential Functions, Growth, and Decay 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. An asymptote is a line that a graphed function approaches as the value of x gets very large or very small.

Holt Algebra Exponential Functions, Growth, and Decay 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.

Holt Algebra Exponential Functions, Growth, and Decay 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!

Holt Algebra Exponential Functions, Growth, and Decay Tell whether the function shows growth or decay. Then graph. Example 1A: Graphing Exponential Functions Step 1 Find the value of the base. The base,,is less than 1. This is an exponential decay function.

Holt Algebra Exponential Functions, Growth, and Decay Example 1A Continued Step 2 Graph the function by using a table of values. x f(x)

Holt Algebra Exponential Functions, Growth, and Decay Tell whether the function shows growth or decay. Then graph. Example 1B: Graphing Exponential Functions 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

Holt Algebra Exponential Functions, Growth, and Decay Tell whether the function p(x) = 5(1.2 x ) shows growth or decay. Then graph. Step 1 Find the value of the base. Example 1C p(x) = 5(1.2 x ) The base, 1.2, is greater than 1. This is an exponential growth function.

Holt Algebra Exponential Functions, Growth, and Decay Step 2 Graph the function by using a table of values. x–12–8– f(x) Example 1C Continued

Holt Algebra Exponential Functions, Growth, and Decay 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.

Holt Algebra Exponential Functions, Growth, and Decay Clara invests $5000 in an account that pays 6.25% interest per year. What will her investment be worth in seven years? Example 2A: Economics Application 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.

Holt Algebra Exponential Functions, Growth, and Decay Clara invests $5000 in an account that pays 6.25% interest per year. What will her investment be worth in seven years? Example 2A: Economics Application Step 2 Use function to evaluate at t = 7. f(t) = 5000(1.0625) t Substitute 7 for t.

Holt Algebra Exponential Functions, Growth, and Decay 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. 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. Example 2B

Holt Algebra Exponential Functions, Growth, and Decay A city population, which was initially 15,500, has been dropping 3% a year. Write an exponential function. What will the population be after ten years? Example 3: Depreciation Application 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. f(t) = 15,500(0.97) 10 =11,430 t=10 for ten years.

Holt Algebra Exponential Functions, Growth, and Decay Notes 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 2. Use a graph to predict the population in ≈ 7.73 billion The value of a $3000 computer decreases about 30% each year. 3. Write a function for the computer’s value. Does the function represent growth or decay? 4. Use a graph to predict the value in 4 years. V(t)≈ 3000(0.7) t ≈ $720.30

Holt Algebra Exponential Functions, Growth, and Decay 5. Tell whether the function shows growth or decay. Then graph. Notes (continued) f(x) = 8(½) x