Exponential Functions and Their Graphs (Section 3-1)

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1 Exponential Functions and Their Graphs (Section 3-1) Essential Question: How do you graph an exponential function? Students will write a summary describing the steps for graphing an exponential function.

2

3 Use a calculator to evaluate a function at the indicated value of x. Round result to 3 decimals places. x Example 1 f ( x) 2, x 3. 1

4 Use a calculator to evaluate a function at the indicated value of x. Round result to 3 decimals places. Example 2 f ( x) 2 x, x

5 Use a calculator to evaluate a function at the indicated value of x. Round result to 3 decimals places. Example 3 x f ( x) 0.6, x 3 2

6 Graph the exponential function by hand. Identify any asymptotes and intercepts and determine whether the graph of the function is increasing or decreasing. Example 4 f ( x) 3 x

7 Graph the exponential function by hand. Identify any asymptotes and intercepts and determine whether the graph of the function is increasing or decreasing. Example 5 f ( x) 4 x

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9 To graph an exponential equation by hand: 1.Fill out a table of values include five points: some bigger than zero and some smaller than zero 2.Plot the points in the table 3.Connect the points with a curve. Essential Question: How do you graph an exponential function?

10 Transformations If you add in the exponent ( y = 2 x+n )your graph shifts to the left n units If you subtract in the exponent ( y = 2 x-n ) your graph shifts to the right n units If you add not in the exponent ( y = 2 x +n) your graph shifts up n units If you subtract not in the exponent ( y = 2 x -n)your graph shifts down n units If you put a negative in front of the equation ( y = -2 x )you reflect across the x-axis If you put a negative in front of the x in the exponent ( y = 2 -x ) you reflect across the y-axis. Essential Question: How do you graph an exponential function?

11 Example 6 Use the graph of y = 4 x to match the function with its graph. y=4 x f(x) = 4 -x g(x) = 4 x+3 h(x) = 4x - 2 A B C

12 Example 7 Use the graph of f(x) = 3 x to describe the transformation that yields the graph of g. a) g(x) = 3 x+1 b) g(x) = 3 x 2 c) g(x) = -3 x d) g(x) = 3 -x

13 Natural Base e: e = Be sure that you understand that in e x e is a number and x is the exponent. Approximating e: e can be approximated by Essential Question: How do you graph an exponential function?

14 Use a calculator to evaluate the function at the indicated value of x. Round your answer 3 decimal places. (nearest thousandth) x Example 8 f ( x) e, x 2

15 Use a calculator to evaluate the function at the indicated value of x. Round your answer 3 decimal places. (nearest thousandth) x Example 9 f ( x) e, x 0. 4

16 Use a calculator to evaluate the function at the indicated value of x. Round your answer 3 decimal places. (nearest thousandth) 2x Example 10 f ( x) 20e, x 0. 01

17 To sketch the graph of an exponential function: 1. Use the calculator to make a table. 2. Plot the points. 3. Connect the points with a smooth curve.

18 Use a graphing utility to construct a table of values for the function. The sketch the graph of the function. Identify any asymptotes. 2 Example 11 y 4 x

19 Use a graphing utility to construct a table of values for the function. The sketch the graph of the function. Identify any asymptotes. Example 12 y e x

20 HW #81 pg (1-43 odd) Essential Question: How do you graph an exponential function?

21 Use a graphing utility to find the point(s) of intersection, if any, of the graphs of the function. Example 13 y y 100e ,500 x

22 (a) Use a graphing utility to graph the function, (b) use the graph to find the open intervals on which the function is increasing and decreasing and (c) approximate any relative max or min values. Example 14 2 x 1 f ( x) 2x e

23 One of the most common places that e is used in everyday applications is in compound interest. Formulas for Compound Interest If interest is compounded times per year the amount in an account is If interest is compounded the amount in an account is Pg 190

24 Example 15 (pg 191) A total of $9000 is invested at an annual interest rate of 2.5% compounded annually. Find the balance in the account after 5 years. Essential Question: How do you graph an exponential function?

25 Example 16 A total of $4000 is invested at an annual interest rate of 5.25% compounded monthly. Find the balance in the account after 6 years. Essential Question: How do you graph an exponential function?

26 Example 17 (pg 191) A total of $12,000 is invested at an annual interest rate of 3% compounded annually. Find the balance after 4 years if the interest is compounded (a) quarterly and (b) continuously. Essential Question: How do you graph an exponential function?

27 Example 18 (pg 192) Essential Question: How do you graph an exponential function?

28 Example 19 (pg 192) Essential Question: How do you graph an exponential function?

29 HW #82 pg (45-53 odd, 57, 63, 65, 67, 69) Essential Question: How do you graph an exponential function?

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