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1 Objective 8 Quadratic Functions The simplest quadratic function is f() = 2. Objective 8b Quadratic Functions in (h, k) form Appling all of Obj 4 (reflections and translations) to the function. f() = a( h) 2 +k verte a > 0 parabola opens a < 0 parabola opens Objective 8a Quadratic Functions in Standard form f() = a 2 +b+c What s the verte? We could and put it in (h,k) form. Good news: f() = a 2 +b+c verte a > 0 parabola opens a < 0 parabola opens For either quadratic form: To find -intercepts, let solve for. To find -intercepts, let solve for. Sometimes we ask How man -intercepts are there? For Obj 8a, You can use the For Obj 8b, Just

2 Objective 8c Ma/Min of Quadratic Function Ob 8a eample The information included in this eample would be asked in separate on line problems. f() = Opens Up/Down -coordinate of verte = How man -intercepts? Ma/Min is Ma/Min is at = Find all intercepts. (For on line problems: Enter them in an order separated b a comma.) Which of the following most closel resembles the graph of f() = c? 2

3 Ob 8b eample The information included in this eample would be asked in separate on line problems. f() = a(+) 2 8,a > 0 Opens Up/Down verte = How man -intercepts? Ma/Min is Ma/Min is at = Find all intercepts for f() = 2( + ) 2 8. (For on line problems: Enter them in an order separated b a comma.) Find all intercepts for f() = 2(2 + ) 2 0. (For on line problems: Enter them in an order separated b a comma.) Which of the following most closel resembles the graph of f() = a(+) 2 2,a > 0? 3

4 Ob 8c eample Studies have found that the relationship between advertising dollars, a, in thousands, and revenue, R, can be modeled b a quadratic function. If R(a) = 4a a , how man thousands of advertising dollars should be spent in order to maimize revenue? (Enter number answer - integers or eact decimals; mathematical operators are not allowed. For eample, 5/2 must be entered as 7.5. Don t tpe an dollar signs, commas, or units. The function given does not represent the results of an actual stud.) Ob 8c eample A large swimming pool is treated regularl to control the growth of harmful bacteria. If the concentration of bacteria, C (per cubic centimeter), t das after treatment, is given b C(t) = 0.4t 2 4.4t+30., What is the minimum concentration of bacteria? (Same cautions as in previous eample.) 4

5 Objective 9 Power Functions f() = n, where n is an integer, n 2 The power functions are classified into 2 groups: f() = n, where n is an even positive integer, n 2 For eample: f() = 2, f() = 4, f() = 6,... f() = 58,... f() = n, where n is an odd positive integer, n 3 For eample: f() = 3, f() = 5, f() = 7,... f() = 59,... 5

6 Objective 20 Solving Polnomial and Rational Inequalities Solve Solve. ( 2 +)(4 2 )(+) 2 < 0 Solve. 3 (+) 3 0 6

7 Solve. 5 > 3 4 Solve Objective 20c How man partitioning points would be needed to solve Recall Obj 0c. Give the domain for each. f() = 3 2 f() =

8 Objective 20d Give the domain. f() = f() = f() = 3 f() = f() = f() =

9 Objective 2 Inverse Functions Illustrate the idea of inverse functions. f() = 2 + f() = Two one-to-one functions are inverses of each other if (f g)() = of g, and (g f)() = for all in the domain of f. for all in the domain We write f to denote the inverse function. Objective 2b How are the graphs of f and f related? If (a,b) is on the graph of = f(), then is on the graph of = f (). 9

10 Objective 2b Eample Select the graph of = f (). A function can be its own inverse. Consider Objective 2a Does ever function have an inverse? to be be the graph of a one- Graph of a function must pass the to-one function. Which are one-to-one functions? {(,2),(,3),(5,4)} {(,2),(3,2),(4,5)} {(,2),(3,3),(4,5)} 0

11 If a function is not one-to-one, restrict the domain in order to define an inverse function. (Recall intro to Obj 2.) Objective 2d Given a function, find the function rule for f. f() = 3 5 Find the function rule for f for f() = (4 ) 5 +7

12 Find the function rule for f for f() = + 3 Find the function rule for f for f() = Objective 22 Eponential Functions. f() = a, a > 0, a Don t allow base to be negative because could be Don t allow base to be because for some. i.e., graph would be linear, not eponential. What s the domain? All reals? If so, we have to define what s meant b irrational eponents. For eample: 4 2 or 4 π We haven t worked with irrational eponents. Good News: The limiting processes of calculus guarantee that irrational eponents are defined, and line up as we want. (See eample below.) 2

13 The eponential functions are classified into 2 groups, depending on the base. f() = a, a > f() = a, 0 < a < We will consider two specific cases to develop the concept. This is not an on line problem eample; ou will not be making tables of values - ou will not be plotting points. ( Consider f() = 4 Consider f() = 4) for an eample of a > for an eample of 0 < a < π 4 50 Objective 22a Properties and Graphs of Eponential Functions f() = a, a > f() = a, 0 < a < 3

14 Objective 22b Graphing Eponential Functions with Reflections or Translations Don t. Don t. Use Obj 4! Select the graph that best represents the graph of each of the following. ( f() = 5 f() = 4) Which function best describes the graph shown? Which function best describes the graph shown? f() = (2.5) f() = (2.5) f() = (2.5) f() = (2.5) f() = (0.4) f() = (0.4) f() = (0.4) f() = (0.4) More Objective 22b Graphing Eponential Functions with Translations Don t. Don t. Use Obj 4! 4

15 Select the graph that best represents the graph of each of the following. f() = 4 3 f() = ( ) +2 5 Which function best describes the graph shown? Which function best describes the graph shown? f() = 6 +3 f() = 6 +3 f() = ( ) +2 5 f() = ( ) f() = (0.6) +3 f() = (0.6) +3 f() = ( ) 2 2 f() = ( ) Objective 22c The eponential function is f() = e because of so man areas of application. ( e e = lim n + ) n n Graph f() = e 5

16 Evaluate e on a scientific calculator (the Windows PC calculator in lab class). Strontium 90 is a radioactive material that decas over time. The amount, A, in grams of Strontium 90 remaining in a certain sample can be approimated with the function A(t) = 225e 0.037t, where t is the number of ears from now. How man grams of Strontium 90 will be remaining in this sample after 7 ears? Objective 22d Solving eponential equations when we can obtain the same base. Eponential functions are one-to-one; that means: if and onl if Rewrite each side (if needed) in terms of a common base; use the smallest base possible. Be sure to replace equals. Solve 5 2+ = 25 3 ( 4 Solve = 9) ( ) 27 8 Objective 23 Logarithmic Functions Consider an eponential function = a What s the inverse function? There is no algebraic operation to solve for. We must define a new function. = log a 6

17 Objective 23a Evaluate Logarithmic Functions log 2 8 = log 25 5 = log /6 2 = log 2 2 = log 2 = Which are defined? (Be careful, sometimes ask Which are undefined? ) log /2 log /4 4 log /2 ( 4) log /2 0 Objective 23b Properties and Graphs of Logarithmic Functions f() = log a, a > 0, a The logarithmic functions are classified into two groups comparable to the eponential functions. Recall Obj 22a = a, a > = a, 0 < a < = log a, a > = log a, 0 < a < 7

18 Objective 23c Graphing Logarithmic Functions with Reflections or Translations Don t. Don t. Use Obj 4! Select the graph that best represents the graph of each of the following. f() = log 4 f() = log /4 ( ) Which function best describes the graph shown? Which function best describes the graph shown? f() = log (5/2) () f() = log (5/2) ( ) f() = log (2/5) () f() = log (2/5) ( ) f() = log (5/2) () f() = log (5/2) ( ) f() = log (2/5) () f() = log (2/5) ( ) More Objective 23c Graphing Logarithmic Functions with Translations Don t. Don t. Use Obj 4! 8

19 Select the graph that best represents the graph of each of the following. f() = log 3 ()+2 f() = log /3 (+2) Which function best describes the graph shown? Which function best describes the graph shown? f() = log (5/2) ()+2 f() = log (5/2) () 2 f() = log (2/5) ()+2 f() = log (2/5) () 2 f() = log (5/2) (+2) f() = log (5/2) ( 2) f() = log (2/5) (+2) f() = log (2/5) ( 2) Objective 23d Domain of Logarithmic Functions (not b graphing) Give the domain. f() = log b (4 5) f() = 5 log b (3) 9

20 ( ) + f() = log b 3 Objective 24a Properties of Logarithmic Functions As used below: a > 0, a, b > 0, b, M > 0, N > 0, > 0, and r represent an real number Definition means Common Logarithms are logarithms base 0; we write instead of. Natural Logarithms are logarithms base e; we write instead of. Objective 24a Eample Which of the following is equivalent to ln5 =? A) 5 e = B) e = 5 C) 5 = e more Properties of Logarithms Product Rule log b (MN) = Must Note: log b (MN) Must Note: log b (M +N) Quotient Rule ( ) M log b = N ( ) M Must Note: log b N Must Note: log b (M N) Power Rule log b M r = 20

21 When Base and Result Match log b b = When Result is log b = Inverse Function Properties Obj 24b Recall Obj 2: (f f )() = and (f f)() = a log a M = log a a r = Objective 24b Eamples Simplif 5 log 5 (3) = log 3 3 5t2 = Objective 24a Eample Which of the following is equivalent to log b ( )? ( ) A) log b C) both A and B B) log b log b D) none is equivalent Objective 24c Eamples Epand using log properties. log b ( 2 z ) ( 2 ) Epand using log properties. log b z(w +3) 2

22 Which of the following is equivalent to A) 2log b +log b log b z +log b (w+3) B) 2log b +log b log b z log b (w+3) C) A and B are the same log b ( 2 ) log b (z(w +3)) another Objective 24c Eample Write as a single logarithm 2log b log b + log 2 bz A) log b 2 z B) log b 2 z Copright c 200, Annette Blackwelder, all rights are reserved. Reproduction or distribution of these notes is strictl forbidden. 22

is on the graph of y = f 1 (x).

is on the graph of y = f 1 (x). Objective 2 Inverse Functions Illustrate the idea of inverse functions. f() = 2 + f() = Two one-to-one functions are inverses of each other if (f g)() = of g, and (g f)() = for all in the domain of f.