Attributes and Transformations of f(x) = e x VOCABULARY

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1 - Attributes and Transformations of f() = e TEKS FOCUS TEKS ()(A) Determine the effects on the ke attributes on the graphs of f() = b and f() = log b () where b is,, and e when f() is replaced b af(), f() + d, and f( - c) for specific positive and negative real values of a, c, and d. TEKS ()(E) Create and use representations to organize, record, and communicate mathematical ideas. VOCABULARY Natural base eponential function A natural base eponential function is an eponential function with base e. Representation a wa to displa or describe information. You can use a representation to present mathematical ideas and data. Additional TEKS ()(D), ()(A), ()(I) ESSENTIAL UNDERSTANDING Eponential functions with base e have the same properties as other eponential functions. Ke Concept The Number e Up to this point ou have worked with rational bases. However, eponential functions can have irrational bases as well. One important irrational base is the number e. The graph of = + has an asmptote at = e or.. A B... As approaches infinit the graph approaches the value of e. Natural base eponential functions are eponential functions with base e. These functions are useful for describing continuous growth or deca. Eponential functions with base e have the same attributes as other eponential functions. Ke Concept Continuousl Compounded Interest amount in account at time t interest rate (annual) A(t) P e rt Principal time in ears Lesson - Attributes and Transformations of f() = e

2 Problem Evaluating e How can ou use a graphing calculator to evaluate e? After ou press the e ke, what kes should ou press? Press, ), and enter. Method Use the e ke. e^(). Method Use the graph of = e. Y=e^(X) X Y. Method Use a table of values for = e. X Y Y=. e. Problem TEKS Process Standard ()(E) Analzing the Attributes of = e Graph the function f () = e and analze the domain, range, intercepts, asmptotes, minimum and maimum on the interval [,.]. Make a table of values and graph the function. f()..... How do ou know the -ais is an asmptote? As decreases, the value of e is positive and gets closer and closer to, but is never equal to. The domain is all real numbers. The range is. The -intercept is (, ). There is no -intercept. The asmptote is the -ais or the line =. The minimum value of f () = e on [ -,.] is f (-) = e -.. The maimum value of f () = e on [ -,.] is f (.) = e O PearsonTEXAS.com

3 Problem TEKS Process Standard ()(D) Analzing = af () for f () = e Graph each function on the same set of aes as the parent function f () = e. What is the effect of the transformation on the range? A =. ~ e Since a =. and a, =. # e compresses the graph of f () b the factor.. B = ~ e Since a =- and a, =- # e reflects the graph of f () across the -ais and stretches it b the factor. How can ou choose an appropriate scale for the -ais? The function = e grows ver quickl as increases. Choose a scale for the -ais that allows ou to show the graph sharpl curving upward. = e =. e - - O The range remains. = e - - O = e - - The range is now instead of. Problem TEKS Process Standard ()(D) Analzing = f () + d for f () = e Graph each function on the same set of aes as the parent function f () = e. What is the effect of the transformation on the range? A = e + Since d = and d, = e + shifts the graph of f () up b units. B = e Since d =- and d, = e - shifts the graph of f () down b units. How can ou check that the graphs are reasonable? Check that the graph of = e + crosses the -ais units above the point where the graph of = e crosses the -ais. = e + = e - - O = e - - O - = e The range is now, instead of. The range is now - instead of. - Lesson - Attributes and Transformations of f() = e

4 Problem Analzing = f ( c) for f () = e What is the value of c? Write the eponent in the form - c. The function = e (+) is = e (-(-)) so c =-. Graph each function on the same set of aes as the parent function f () = e. What is the effect of the transformation on the asmptote? A = e (+) B = e ( ) Since c =- and c, = e (+) shifts the graph of f () to the left b - or units. = e ( + ) O = e Since c = and c, = e (-) shifts the graph of f () to the right b units. = e O = e ( ) The asmptote remains =. The asmptote remains =. Problem Continuousl Compounded Interest What is the unknown? The unknown is the amount A in the account after ears. Scholarships Suppose ou won a contest at the start of th grade that deposited $ in an account that pas % annual interest compounded continuousl. How much will ou have in the account when ou enter high school ears later? Epress the answer to the nearest dollar. A = P # e rt = e (.)() Substitute values for P, r, and t. = e. Simplif. Use a calculator. Round to the nearest dollar. The amount in the account, to the nearest dollar, is $. Write our answer,, in the grid. PearsonTEXAS.com

5 ONLINE H O M E W O R K PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd tutorial video. For additional support when completing our homework, go to PearsonTEXAS.com. Graph each transformation of the parent function f () = e on the same set of aes as the parent function. Analze the effect of the transformation on the graph of the parent function.. = # e. = e +. = e (-). =- # e. = e -. = e (+) Analze the minimum and maimum value of the function f () = e on the given interval.. [ -, ]. [, ]. [ -., ]. [ -, -.] Write the domain and range of each function using interval notation. Then analze the intercepts and asmptotes.. = e +. = e (-.). = e -. = # e +. Analze Mathematical Relationships ()(F) Compare the domain, range, intercepts, and asmptotes of the function =-e with those of the parent function = e.. You invest $ in an account that pas % annual interest compounded continuousl. a. Write a function that models the amount in the account after ears. b. Write the domain and range in inequalities for the function in this situation. c. How much will ou have in the account after ears? Match each function with the correct graph.. = # e (-). = # e -. = # e - A. B. C. - O - - O - - O -. Use Multiple Representations to Communicate Mathematical Ideas ()(D) Use an inequalit to write the range of the function = e (-) +. Then use a table and a graph to eplain how ou know our answer is correct. Lesson - Attributes and Transformations of f() = e

6 Determine whether each statement is alwas, sometimes, or never true.. The maimum value of the function = e on the interval [a, b] occurs at = b.. The function = e (-c) has the same asmptote as the parent function = e.. The graph of = e + d, d, intersects the graph of the parent function = e.. If a, then the graph of = ae lies in the third and fourth quadrants.. For an real number k, the graph of = e intersects the vertical line = k.. The function = e models the average number of dail visitors to a website, in thousands, ears after. a. What is the -intercept and what does it represent? b. What was the increase in the average number of dail visitors from to? c. Is the increase in the average number of dail visitors over an -ear period the same as our answer to Part b? Eplain. TEXAS Test Practice. Which function could be shown in the graph? A. =-e (-) C. =-e - B. =-e + D. =-e (+). Which of the following is a true statement about the functions = e and = e (-)? F. Both functions have the same -intercept. G. Both functions have one -intercept. H. The graph of one function is a vertical translation of the graph of the other function. J. The range of both functions is.. A student uses software to draw the graph of = e. Then she uses the software to reflect the graph across the -ais and translate it units down. The resulting graph is the graph of which function? A. =-e - B. =-e (-) C. =-e (+) D. = e (-) -. On which interval is the maimum value of = e less than e? F. [, ] G. [ -, ] H. [ -, ] J. [, ]. Give an eample of a function whose parent function is = e and whose graph has an -intercept at (, ). Then graph the function.. Use an inequalit to write the range of the function = ae (-c) + d for a and for a. O PearsonTEXAS.com