7.1 Exponential Functions, Growth, and Decay

7.1 Exponential Functions, Growth, and Decay

7.1 EXPONENTIAL FUNCTIONS, GROWTH, AND DECAY Objective Write and evaluate exponential expressions to model growth and decay situations. Vocabulary exponential function base asymptote exponential growth exponential decay Remember! Negative exponents indicate a reciprocal. For example: Warm Up Evaluate. 1. 100(1.08)20 2. 100(1 0.02)10

466.1 81.71 3. 100(1 + 0.08)10 46.32 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 graph of the parent function f(x) = 2x is shown. Why not f(x) = 1x as the parent function? Domain: Range:

Look at the graph. What questions come to mind? A function of the form f(x) = abx, 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. Example 1A: Graphing Exponential Functions Tell whether the function shows growth or decay. Then graph. 3 f x 10 4 x Step 1 Find the value of the base.

Step 2 Graph the function by using a table of values. x -1 0 2 4 6 f(x) Highlight your point on the graph

8 10 12 Check It Out! Example 1 Tell whether the function p(x) = 5(1.2x) shows growth or decay. Then graph. Step 1 Find the value of the base. Step 2 Graph the function by using a table of values. x f(x) 12

8 4 Highlight your point on the graph 0 4 8 10 You can model growth or decay by a constant percent increase or decrease with the following formula:

Questions: In the formula, the base of the exponential expression, 1 + r, is called the growth factor. Similarly, 1 r is the decay factor. Example 2: Economics Application 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. We will learn algebra techniques for solving. Until then we use the graph or table to solve the equation Example 3: Depreciation Application 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 8000. f(t) = a(1 r)t Exponential decay function. 10,000 0 150

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