a > 0 parabola opens a < 0 parabola opens

Transcription

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 *************** Plan of Attack. Factor, if needed. (Watch for Diff of Squares and Factoring out GCF.) 2. Find Partitioning Points. These are points that make the epression (from factors in the numerator) or make the epression (from factors in the denominator). Set each factor equal to 0 to find these Partitioning Points. 3. Mark these on a number line (must be in number-line order). 4. Make a Sign-Chart. Select a value in each interval that s created b these partitioning points (don t use one of end-points). Plug this value into each separate factor and record whether the result is + or. Consolidate the signs from all the factors. *************** Solve Solve. ( 2 +)(4 2 )(+) 2 < 0 6

7 Solve. ( 2 +36) 2 ( )(4 ) < 0 Solve. 3 (+) 3 0 Solve. ( 30)( 2) (00 )(3 5) > 0 Solve. 5 > 3 4 Solve

8 Objective 20c How man partitioning points would be needed to solve? *************** Plan of Attack Obtain a single fraction on one side with 0 on the other. That is, make the problem read to be solved b the sign-chart method. NOTE: You can t or multipl b an epression that contains the variable because ou won t know if the inequalit sign should be reversed. NOTE: If the denominators are constants, then ou are allowed to cross-multipl (multipl b LCM). *************** > < 4 8

9 Objective 20d Find the domain when a sign-chart is needed. Recall Obj 0c. Give the domain for each. f() = 3 2 f() = f() = 3 2 Objective 20d Give the domain. f() = f() = f() = 3 f() =

10 These are special cases ou ma see in our Practice (or Hmwk Quiz) problems, but won t encounter in the Lab Quiz or Test problems. We need to be aware that an even-root radical can have Domain all Reals. f() = f() = 2 25 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. 0

11 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 (). Objective 2b Eample Select the graph of = f (). A function can be its own inverse. Consider

12 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)} If a function is not one-to-one, restrict the domain in order to define an inverse function. (Recall intro to Obj 2.) 2

13 Objective 2d Given a function, find the function rule for f. *************** Plan of Attack. Write for f() (to simplif the notation). 2. Solve for. For applied mathematicians, when units are usuall associated with the variables, ou have the inverse function. 3. For our College Algebra course, we will interchange and to write the inverse as a function of. 4. Write f () for (to return to function notation). *************** Find the function rule for f for f() = 3 5 Find the function rule for f for f() = (4 ) 5 +7 Find the function rule for f for f() =

14 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 Does this define a function? 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.) 4

15 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

16 Objective 22a Properties and Graphs of Eponential Functions f() = a, a > f() = a, 0 < a < 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) 6

17 More Objective 22b Graphing Eponential Functions with Translations Don t. Don t. Use Obj 4! 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() = ( )

18 Objective 22c The eponential function is f() = e because of so man areas of application. ( e e = lim n + ) n n Graph f() = e Evaluate e on a scientific calculator (the Mac 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? $8,000 is invested in a bond trust that earns 5.9% interest compounded continuousl. The account balance t ears later can be found with the function A = 8000e 0.059t. How much mone will be in the account after 6 ears? 8

19 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 4 Solve = 9 ( ) Solve ( ) 4 25 =

20 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 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 20

21 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 < 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 ( ) 2

22 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! Select the graph that best represents the graph of each of the following. f() = log 3 ()+2 f() = log /3 (+2) 22

23 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) ( ) + f() = log b 3 f() = log 3 (4 2 ) f() = log 3 ( 2 +4) 23

24 Objective 24 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 - Obj 24a 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 Properties of Logarithms - Obj 24b 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 = When Base and Result Match log b b = When Result is log b = 24

25 Inverse Function Properties - Obj 24c Recall Obj 2: (f f )() = and (f f)() = a log a M = log a a r = Objective 24c Eamples Solve for if 5 log 5 (3) = 5 Solve for if lne 5 = 3 Objective 24b 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 Appling Log Properties - Objective ( ) 24d 2 Epand using log properties. log b z ( 2 ) Epand using log properties. log b z(w +3) 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)) 25

26 another Objective 24d Eample Write as a single logarithm 2log b log b + log 2 bz A) log b 2 z B) log b 2 z another Objective 24d Eample Write as a single logarithm. 2log b (z w) log b w +3log b z +log b log b (+w) another Objective 24d Eample If log b 2 = l and log b 5 = m, epress log b 00 in terms of l and m. another Objective 24d Eample If log b 2 = l and log b 5 = m, epress (log b 4) (log b 25) in terms of l and m. Copright c 200-present, Annette Blackwelder, all rights are reserved. Reproduction or distribution of these notes is strictl forbidden. 26