LEARNING GOALS – LESSON 7.1 WARM-UP Evaluate the following expressions: 2-3 2. 50 3. 10(1-.75)2 Find the values indicated using the graph: f(2) = _____ 5. f(___) = 4 LEARNING GOALS – LESSON 7.1 Write and evaluate exponential expressions to model growth and decay situations. In case of a school closing, the principal calls 3 teachers. Each of these teachers calls 3 other teachers and so on. How many teachers will have been called in the 4th round of calls? Draw a diagram to show the number of Teachers called in each round of calls. _____________ _____________ _____________ * Notice that after each round of calls the number of families contacted is a power of 3. Write a function to represent this situation where y is the number of people contacted and x is the round number.

What does the following phrase mean in your own words? “The number of transistors required per circuit has increased exponentially since 1965.” Check out the y-values. What pattern do you notice? Growth that ______________ every year can be modeled by an _______________ function. The parent exponential function: f(x) = bx, where x is the variable. When something doubles the b value will be ______. What about when it triples?

Exponential Growth Exponential Decay The graph of the parent function: f(x)= 2x is shown. Domain : { R } all _______ numbers Range : {y|y > 0} all ________ numbers The function never reaches the x-axis because the value of 2x≠0 ever. In this case, the x-axis is an ____________________. Exponential Growth Exponential Decay f(x) = abx a > 0 (_____________) b > 1 (_________ than 1) 0 < b < 1 (____________ 0 and 1) As x increases y ________________ As x increases, y ________________

Example 1A: Graphing Exponential Functions Tell whether the function shows growth or decay. Then graph. Step 1 Find the value of the base. Determine if it represents growth or decay. Step 2 Graph the function by using a table of values. Determine appropriate x values. Plug function into graphing calculator and use table feature to fill in chart. Graph. x f(x) Check by graphing in calculator. Be sure to appropriately represent asymptotes!

Exponential Growth/Decay (at a constant rate) Example 1B: Graphing Exponential Functions Tell whether the function shows growth or decay. Then graph. g(x) = 100(1.05)x Step 1 Is this GROWTH or DECAY ? (Circle One) Step 2 PREPARE TO GRAPH the function by making a table of values. x f(x) Exponential Growth/Decay (at a constant rate) What do you think the ± tells us about the formula? + means ____________ and – means ______________

Example 2: Biology Application In 1981, the Australian killer whale population was 350 and increased at a rate of 14% each year since then. Use a graph to predict when the population will reach 20,000. Step 1: Determine if this problem is representing a growth or decay; write the general formula. Step 2: Substitute in the info that you have. Step 3: Since you don’t know how to solve for t yet (because it is an exponent) graph it and using the [TRACE] feature estimate what value of t (x) will give you 20,000 (y). Step 4: Be sure to use the t value that you found to count years past 1981 to give the year the population will reach 20,000.

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. Step 1 Determine if this problem is representing a growth or decay; write the general formula. Step 2 Substitute in the information that you have. Step 3 Since you don’t know how to solve for t yet (because it is an exponent) graph it and using the [TRACE] feature estimate what value of t (x) will give you a number just below 8,000 (y).