1 Section 0.3 Power and eponential functions (5/6/07) Overview: As we will see in later chapters, man mathematical models use power functions = n and eponential functions =. The definitions and asic properties of these functions, which are studied in precalculus courses, are reviewed in this section. We will discuss applications of these functions in later chapters, eginning with Chapter, where we stud their derivatives. We also descrie here how formulas for functions can e modified to translate, reflect, epand, and contract their graphs, The section closes with notes on the histor of analtic geometr. Topics: Power functions = n and their graphs Vertical and horizontal translation Reflection, magnification, and contraction Eponential functions = and their graphs Laws of eponents Historical notes Power functions A power function is a function of the form = n, where is the variale and n is a constant. If n is a positive integer, then n equals the product of n s, as in the formula 3 =. If n is a positive fraction, p/q, then n = p/q is the qth root of the pth power of, which also equals the pth power of the qth root of. In the case of n = 3 5, for eample, we have n = 5/3 = 3 5 = [ 3 ] 5. If n is a negative integer or fraction, so that n = m with m a positive integer or fraction, then n equals / m, as in the formulas 3 = 3 = 5/3 = 5/3 = 3 5 = [ 3 ] 5. all. To have formulas and identities involving n appl with zero eponents, 0 is defined to e for If n is not an integer or a fraction, it is irrational and has an infinite decimal epansion, which is used to define n for positive. The irrational numer, for eample, has the decimal epansion = , and if we want to define 0, we let n =. e the numer otained taking onl one digit after the decimal point in the epansion of, let n =. e the numer otained taking two digits after the decimal point, and so forth. This gives us an infinite string of rational numers n, n,n 3,... that approaches. We sa that is the limit of the numers n, n,n 3,... The numers 0 n, 0 n, 0 n 3,... are defined ecause the eponents n, n,n 3,... are rational. And, just as the numers n,n, n 3,... approach their limit, the numers 0 n,0 n,0 n 3,... approach their limit, which is defined to e 0. The first seven of the numers 0 n j are calculated in the net question. We will discuss limits in Chapter.
2 p. (5/6/07) Section 0.3, Power and eponential functions Question The following tale gives the first seven of numers n, n,n 3,... that approach = Use a calculator or computer to complete the second row of include approimate decimal values of 0 n, 0 n, 0 n 3,...,0 n 7. j 3 n j n j. = The same procedure is used to define n for an irrational n and positive. We let n j denote the rational numer otained taking j digits after the decimal point in the decimal epansion of n. Then the numers n j approach n and the numers n j approach n. The epression n is defined unless it involves dividing zero, taking an even root of a negative numer, or taking an irrational power of a negative numer. Consequentl, = n is defined for all with three eceptions: it is not defined at 0 if n is negative; it is not defined for negative if n is a fraction with an even denominator, and it is not defined for negative is n is irrational. The graphs of = n Figures through show four curves = n with odd integers n. The curve = with n = in Figure is a line through the origin. The shapes of the other curves are analzed in the following questions. = FIGURE Question (a) Wh does = 3 in Figure pass through the origin and wh is it aove the -ais for > 0 and elow the -ais for < 0? () Use the formula 3 = to eplain wh = 3 is much closer to the -ais than = for nonzero ver close to 0 and is much farther from the -ais than = for ver far from 0. = 3 = = /3 = = FIGURE FIGURE 3 FIGURE Notice that the curve = 3 in Figure gets steeper as moves awa from 0.
3 Section 0.3, Power and eponential functions p. 3 (5/6/07) Question 3 Question (a) Wh does = /3 in Figure 3 pass through the origin and wh is it aove the -ais for > 0 and elow the -ais for < 0? () Use the formula = /3 /3 to eplain wh = is closer to the -ais than = /3 for < < 0 and 0 < < and is farther from the -ais than = /3 for < and >. Notice that the curve = /3 in Figure 3 gets less steep as moves awa from 0. Eplain wh = in Figure (a) does not intersect the -ais and is aove it for > 0 and elow it for < 0, () is far from the -ais for small positive and small negative, and (c) is close to the -ais for large positive and large negative. The reasoning in Questions through can e used to show that, for an constant n not equal to 0 or, the portion of = n for positive is similar to the portion of one of the curves in Figures through for positive. For n > (Figure 5), the curve = n, like = 3 in Figure, curves up to the right from the origin. For 0 < n < (Figure 6), the curve = n, like = /3 in Figure 3, rises up from the origin ut gets less steep as moves to the right. For n < 0 (Figure 7), the curve = n, like = in Figure, comes down to the right of the -ais and approaches the -ais as moves to the right. = n (n > ) = n (0 < n < ) = n (n < 0) FIGURE 5 FIGURE 6 FIGURE 7 The nature of the graph = n for < 0 depends on the value of n. There are three possiilities: = n is either an odd function, is an even function, or is not defined for negative. We are using here the following definition. Definition A function = f() is even if f( ) = f() for all in its domain and is odd if f( ) = f() for all in its domain. The graphs of odd functions are smmetric aout the origin; the graphs of even functions are smmetric aout the -ais. The functions = 3, = /3, and = of Figures through are odd ecause ( ) 3 = and ( ) /3 = /3 for all and ( ) = for 0. Their graphs are smmetric aout the origin. The functions =, = /3, and = of Figures 8 through 0 are even ecause ( ) = and ( ) /3 = /3 for all and ( ) = for 0. Their graphs are smmetric aout the -ais. The functions = 5/, = /, and = / of Figures through are not defined for negative ecause the involve the square root / =. Consequentl, their graphs do not etend to the left of the -ais.
4 p. (5/6/07) Section 0.3, Power and eponential functions 3 = = /3 3 = FIGURE 8 FIGURE 9 FIGURE 0 = 5/ 3 = / = / 3 FIGURE FIGURE FIGURE 3 Eample Match the functions (a) =, () = 5, and (c) = / to their graphs in Figures through 6. 3 FIGURE FIGURE 5 FIGURE 6 Solution (a) Here the power n = is a negative numer. Consequentl, = is similar to = for positive. The function = is even, so its graph is smmetric aout the -ais and is in Figure 5. () The power 5 in this case is greater than, so = 5 is similar to = 3 for 0. The function = 5 is odd, so its graph is smmetric aout the origin and is in Figure 6. (c) Because n = is a fraction etween 0 and, = / is similar to = /3 for 0, ut since n = has an even denominator, the function is not defined for 0, and its graph is in Figure.
5 Section 0.3, Power and eponential functions p. 5 (5/6/07) Vertical and horizontal translation If we add a positive constant to a function = f(), we otain the function = f()+k, whose graph is otained from the graph of f raising it k units. Sutracting a positive constant k ields = f() k, whose graph is otained lowering the graph of = f() k units (Figure 7). The raising or lowering of a graph is called vertical translation. = f() + k k k = f() = f() k = f( + k) = f() = f( k) k k FIGURE 7 FIGURE 8 If, on the other hand, we add a positive constant k to the variale in the formula = f(), we otain = f( + k), whose value at 0 k is f (( 0 k) + k) = f( 0 ), which is the value of = f() at 0. Consequentl, = f( + k) is otained shifting the graph of f to the left k units, as shown in Figure 8. This action is called horizontal translation. Sutracting a positive constant k from the variale gives = f( k), whose value at 0 + k is f (( 0 + k) k) = f( 0 ), which is the value of = f() at 0. Hence, = f( k) is the curve = f() shifted k units to the right, as is also shown in Figure 8. Eample Solution Sketch the graph of the function = + 3 completing the square. We complete the square in the formula = + 3 adding and sutracting the square of half the coefficient of. This gives = ( +) +3 or = ( ) +. Its graph in Figure 9 is the curve = translated up units and unit to the right. 6 = ( ) + FIGURE 9 3 Reflection Multipling a function = f() gives the function = f() whose value at is the negative of the value of = f(). Its graph is the mirror image of = f() relative to the -ais. Multipling the variale gives the function = f( ) whose value at is the value of = f() at. Its graph is the mirror image of = f() relative to the -ais (Figure 0). Read this paragraph carefull: = f( + k) is = f() shifted k units to the left, not to the right.
6 p. 6 (5/6/07) Section 0.3, Power and eponential functions = f( ) = f() = f() FIGURE 0 Question 5 Draw the graph of = f( ), where f is the function whose graph is in Figure 0. Eample 3 Solution Sketch the graphs of (a) = and () =. (a) The graph of = in Figure is the mirror image relative to the -ais of the curve = in Figure. () The graph of = in Figure is the mirror image relative to the -ais of =. = = 3 3 FIGURE FIGURE Magnification and contraction If we multipl a function = f() a constant k >, we otain the function = kf(), whose graph is otained from = f() multipling the -coordinate of ever point on it k. This magnifies the curve verticall, as shown the middle and upper curves in Figure 3. Similarl, if we divide the function k >, we otain = f(), whose graph is otained from = f() dividing the -coordinate of k each point k. This contracts the curve verticall, as shown the lower curve in Figure 3. (k > ) = kf() (k > ) = f(k) = f() = f() = k f() = f(/k) FIGURE 3 FIGURE
7 Section 0.3, Power and eponential functions p. 7 (5/6/07) Multipling the variale a constant k > contracts the graph horizontall since = f(k) for positive k has the same value f (k( 0 /k)) = f( 0 ) at 0 /k as = f() has at 0. Similarl, dividing the variale k > magnifies the graph horizontall (Figure ). Eample The curve drawn with a heav line in Figure 5 is the graph of = G(). Is the other curve the graph of = G(), = G( ), = G(), or = G()? = G() FIGURE Solution The second curve in Figure 5 has the equation = G() ecause it is otained magnifing = G() a factor of verticall and contracting it a factor of horizontall. Eponential functions An eponential function is a function of the form =, where the eponent is the variale and is a positive constant, called the ase. All eponential functions are defined and positive for all, and their graphs pass through the point (0, ) since 0 = for an positive. If =, then is the constant function = (Figure 6). If is greater than, then the graph approaches the -ais on the left and curves up on the right (Figure 7). If 0 < <, then the graph approaches the -ais on the right and curves up on the left (Figure 8). 6 = ( = ) = ( > ) 6 6 = (0 < < ) FIGURE 6 FIGURE 7 FIGURE 8 Notice that the function = with positive is neither even or odd; its graph is not smmetric aout the -ais nor aout the origin. Read this paragraph carefull: for k > 0, = f(k) is = f() contracted a factor of k, not epanded a factor of k.
8 p. 8 (5/6/07) Section 0.3, Power and eponential functions Eample 5 Draw the curve = 5 + 3( ). Solution The curve = 5 + 3( ) is = magnified verticall a factor of 3 and then translated up 5 units. It is drawn in Figure 9, where the values (0) = 5 + 3( 0 ) = = 8 and () = 5 + 3( ) = 5 + = 7 on it have een plotted. The curve has = 5 as a horizontal asmptote. = 5 + 3( ) = 5 FIGURE 9 Question 6 Draw the curve = 5 3( ). Eample 6 A sample of radioactive radium-66 has mass M(t) = 6 ( ) t/60 grams at time t (ears). (a) What is the mass of the sample at t = 0, t = 60, and t = 30 ears and how are these numers related? () Draw the graph of M = M(t) in a tm-plane. Solution (a) The formula M(t) = 6 ( ) t/60 gives M(0) = 6 ( ) 0 = 6, M(60) = 6 ( ) = 8, and M(30) = 6 ( ) =. Consequentl, M(60) is half of M(0) and M(30) is half of M(60). (These calculations illustrate the fact that the halflife of radium is 60 ears.) () The graph of M = M(t) in Figure 30 is otained plotting the values from part (a). 6 8 M (grams) M = 6 ( ) t/60 FIGURE t (ears)
9 Section 0.3, Power and eponential functions p. 9 (5/6/07) The natural eponential function As we will see later, the most useful eponential function in calculus is the natural eponential function = e, whose ase is an irrational numer e = that will e defined in Chapter. The graph of = e is shown in Figure 3. = e FIGURE 3 Laws of eponents The following rules for working with eponents are valid for an numers and if is positive. If is negative, the hold for all values of and such that all epressions involved are defined. = + () ( ) = () (c) = c (3) = () The advantage of using eponential notation and these rules is illustrated in the net eample. Eample 7 (a) Simplif the formula = 3 without using fractional or negative eponents taking the sith power of oth sides, simplifing, and taking the sith root. () Simplif = 3 using fractional and negative eponents and rules () through (). Solution (a) Taking sith powers of oth sides of = ( ) 6 6 = ( 3 ) 6 = 3 gives ( )( )( ) ( )( ) = =. Then taking sith roots ields = 6 since > 0. () Fractional eponents enale us to make a more direct calculation: = 3 = / /3 = / /3 = /6 = 6.
10 p. 0 (5/6/07) Section 0.3, Power and eponential functions Historical notes General algeraic equations were first studied in the siteenth centur, ut without the modern convention of using a single letter for the unknown. The Italian phsician and algeraist Gerolamo Cardano (50 57), for eample, used the Latin sentence, Cuus p 6 reus aequalis 0, for the equation that we would write = 0. In Cardano s sentence, the word cuus denotes the cue of the unknown, p stands for plus, reus denotes the unknown, and aequalis means equals. The use of a single letter for the unknown and of eponents for positive integer powers was popularized a treatise La Géométrie, written in 637 the French philosopher Réné Descartes ( ) as an appendi to a work on the philosoph of science. Réné Descartes Pierre Fermat ( ) (60 665) Descartes and a French lawer Pierre Fermat (60 665) are considered the inventors of analtic geometr as a tool for giving geometric meaning to aspects of algera and calculus. Greek mathematicians, including Euclid (ca. 300 BC), Archimedes (87 BC), and Apollonius (ca. 5 BC), used the equivalent of coordinate sstems with the theor of proportions for studing geometric figures, and algera was emploed in the siteenth centur to solve geometric prolems. It was Descartes and Fermat, however, who first studied curves defined equations as well as their geometric properties and who made etensive use of the association etween the algera of equations and the geometr of curves. Fermat s work was not pulished until after his death, more than fort ears after the pulication of Descartes La Géométrie, so Descartes often receives more credit for creating what now is known as analtic or cartesian geometr. Fermat and Decartes generall used onl positive coordinates. Negative coordinates were first used sstematicall Isaac Newton (6 77) in his Enumeration of Curves of Third Degree (676). Fermat s name has een in the news in recent ears ecause his famous last theorem, that n + n = z n has no nonzero integer solutions,, and z for integers n >, has finall een proved. Responses 0.3 Response The tale is completed elow. (Notice that all of the values of 0 n j egin with 5, the last five egin with 5.95, and the last two egin with This illustrates that the numers approach the infinite decimal 5.95 = 0.) j j j. =
11 Section 0.3, Power and eponential functions p. (5/6/07) Response (a) = 3 passes through the origin ecause 0 3 = 0, is aove the -ais for > 0 ecause 3 is positive for > 0, and is elow the -ais for < 0 ecause 3 is negative for < 0. () = 3 is much closer to the -ais than = for ver small nonzero and is much farther from the -ais than = for large positive or negative ecause 3 = equals multiplied a ver small positive numer for small nonzero and multiplied a large positive numer for large positive or negative. Response 3 Response Response 5 (a) = /3 passes through the origin ecause 0 /3 = 0 and is aove the -ais for positive and elow the -ais for negative ecause /3 is positive for positive and negative for negative. () = is closer to the -ais than = /3 for < < 0 and 0 < < and is farther from the -ais than = /3 for < and > ecause = /3 /3 equals /3 multiplied a positive numer less than if < < 0 or 0 < < and equals /3 multiplied a numer greater than if < or >. (a) = does not intersect the -ais ecause = / is not defined at = 0. () = is far from the -ais for ver small close to zero ecause = / is ver large when is ver small. (c) = is close to the -ais if is a large positive or negative numer ecause then = / is ver small. = f( ) is = f() reflected aout the -ais and aout the -ais. Figure R5 = f( ) 5 = 5 3( ) = 5 5 Figure R5 Figure R6 Response 6 = 5 3( ) is = 5 + 3( ) (Figure 9) reflected aout the - and -aes. Figure R6
12 p. (5/6/07) Section 0.3, Power and eponential functions Interactive Eamples 0.3 Interactive solutions are on the we page http// ashenk/.. Solve the equations (a) 3/ = 8 and () /3 = 6 for.. The curve in Figure 3 has the equation = a with constants a and. What are those constants? 8 6 FIGURE Do the two curves in Figure 33 have the equations = 5 + e, = 5 e, = 5 + e, or = 5 e? 5 0 FIGURE 33. Solve the equation 93 9 = 3 for using the fact that if = with a positive, then =. 5. (a) Determine the general shape of the curve = 3 / without generating it on a calculator or computer. () Sketch the curve plotting at least one point on it. 6. (a) Determine the general shape of the curve = + without generating it on a calculator or computer. () Sketch the curve plotting at least one point on it. In the pulished tet the interactive solutions of these eamples will e on an accompaning CD disk which can e run an computer rowser without using an internet connection.
13 Section 0.3, Power and eponential functions p. 3 (5/6/07) 7. Figure 3 shows the graph of a function = P() and the curves = P() and = P() +. Which curve is which? 3 FIGURE The curve drawn with a fine line in Figure 35 is the graph of = H(). (a) Is the other curve aove the -ais the graph of = H() or of = H(/)? () Give equations in terms of H for the two curves elow the -ais. = H() FIGURE 35 Eercises 0.3 A Answer provided. O Outline of solution provided. C Graphing calculator or computer required. CONCEPTS: C. Generate the curves = and = together in the window.5.5, and cop them on our paper. Then use the equation = ( ) to eplain wh = is elow = for some values of and aove it for others. C. Generate the curves = and = together in the window.5.5, and cop them on our paper. Then use the equation = ( ) to eplain wh = is elow = for some values of and aove it for others. 3. Derive the identit = + for a positive constant in the case of = and = 3 writing = and 3 =.. Derive the identit ( ) = for a positive constant in the case of = and = 3 writing ( ) 3 as ( ) ( ) ( ) and then writing for. 5. How can the function z = e either a power function or an eponential function?
14 p. (5/6/07) Section 0.3, Power and eponential functions BASICS: 6. Find all real solutions of (a) = k +, () = k 3, and (c) = k + 0. Here k is a positive constant. 7. Solve the following equations for recognizing powers of and 0 and using the fact that if = with positive, then =. (a) = 8 () = 8 (c) 0 = 0.00 (d) 0 0 = 00 (e) (0 ) = 00 (f) 6 = 6 8. The middle curve in Figure 36 is = 5 =. Which is which? + (/) 5 +. The other curves are = 5 + () and 6 FIGURE 36 In Eercises 9 through solve the equations for. 9. O 3 = 5 0. A 3. 5 = 6 = 0. ( ) 3 =. In Eercises 3 through 9 solve the equations for using the fact that if = with positive, then =. /3 3. O 3 = O 7 = (7 ) 0. 5 = 8 7. A 3 = 9(3 ) ( 5. A ) = 8. ( ) = = 7 What are the domains of the functions in Eeercises 0 through 3? 0. O = /3. A = / 3. = 3/. =
15 Section 0.3, Power and eponential functions p. 5 (5/6/07) Determine the general shapes of the curves in Eercises through 3 without generating them on our calculator or computer analzing their equations. Then sketch them plotting at least one point on each.. A = 5/ 5. = 0 + 5/ 6. O = 5 /( + ) 8. = + ( ) 3 9. O = 30. A = 0 e 7. A = + 3. = 0e /0 3. The lower curve in Figure 37 is = P() and the upper curve is = A + P(B) with constants A and B. What are A and B? 3 = A + P(B) = P() FIGURE Figure 38 shows the curves = L(+k) and = L( k) for a function = L() and a positive constant k. (a) Which is which and what is the value of k? () Draw the graph of = L(). FIGURE A According to Newton s law of gravit, an oject that weighs one pound on the surface of the earth weighs w = 6r pounds when it is r thousand miles from the center of the earth. (The radius of the earth is thousand miles.) Sketch the portion of w = 6r for r in an rw-plane. 35. A A o throws a all straight up in the air at time t = (seconds) and catches it at t =. Because there is no air resistance, the all is h = 6 6t feet aove his hand for t. Sketch the portion of h = 6 6t for t in a th-plane 36. A It costs a factor 5 dollars to manufacture each pint of a chemical, plus an overhead of 00 on each atch produced. The cost for a atch of pints is therefore dollars, and the average cost of a atch of pints is A = = dollars per pint. Draw the portion of A = for > 0 in an A-plane. 37. A culture of acteria contains 500 acteria initialll and, ecause the numer doules ever 3 das, the culture contains N = 500( t/3 ) acteria t das later. Draw the graph of this function in a tn-plane.
16 p. 6 (5/6/07) Section 0.3, Power and eponential functions 38. When air pressure is measured in atmospheres, the air pressure at the surface of the earth is atmosphere. At an altitude h < 80 kilometers aove the surface of the earth, the air pressure is P = ( ) h/5.8. Draw the graph of this function in an hp-plane. EXPLORATION: 0. O (5 )(5 ) = 5 3 for.. A Solve (5 )(5 ) = 5 for.. A The curve = () can e otained from = contracting it horizontall or magnifing it verticall. Eplain. 3. The curve = e + can e otained from = e horizonta translation or vertical magnification. Eplain.. A Which of the curves in Figures 39 through is the graph of = a + / with a > 0? Is positive or negative? Give our reasoning. FIGURE 39 FIGURE 0 FIGURE FIGURE FIGURE 3 FIGURE 5. Which of the curves in Figures 39 through is the graph of = a + 3 for constants a and? Is a positive or negative? Is positive or negative? Give our reasoning. 6. Which of the curves in Figures 39 through is the graph of = a + / for some constants a and? Is a positive or negative? Is positive or negative? Give our reasoning. 7. Which of the curves in Figure 5 has the equation = + a (a) with 0 < a <, () with + 0 < < a, and (c) with a < 0 and > 0? Give our reasoning.
17 Section 0.3, Power and eponential functions p. 7 (5/6/07) I II III FIGURE 5 C 8. Generate = + a on our calculator or computer in the window, 5, first for a = 0, and, and then for a = 0,, and. Eplain how changing a changes the graph and wh. C 9. How does changing change the curve = and wh? (Generate the curve for sample values of in the window, with -scale = 0.) 50. The curve = M() is shown in Figure 6. Draw the curves (a) = M(/), () = M(), (c) = M(), (d) = M()/, (e) = M()+, (f) = M( ), and (g) = M(/). 6 = M() FIGURE Figure 7 shows the graphs of =.5, = e, and = 6. (a) Which is the upper curve, which is the middle curve, and which is the lower curve for > 0? () Which is the upper curve, which is the middle curve, and which is the lower curve for < 0? 3 FIGURE 7 5. Find constants and C such that E() = C has the values in the following tale: 0 3 E()
18 p. 8 (5/6/07) Section 0.3, Power and eponential functions 53. Figure 8 shows the curve = with constant > and the mirror image of this curve aout the origin. What is the equation of the second curve? FIGURE 8 C 5. Generate the curves = and = in the window.5 6,.5 5 to see that the equation = has two positive solutions and one negative solution. Find the positive solutions trial and error and use a calculator or computer to find the approimate value of the negative solution. 55. Give a formula for the surface area A = A(V ) (square meters) of a cue as a function of its volume V (cuic meters) and draw the graph of this function. (End of Section 0.3)
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0_00.qd //0 :5 PM Page 09 SECTION. A Summar of Curve Sketching 09 0 00 Section. 0 0 00 0 Different viewing windows for the graph of f 5 7 0 Figure. 5 A Summar of Curve Sketching Analze and sketch the graph
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(
Objectives C H A P T E R 4 Polnomial functions To be able to use the technique of equating coefficients. To introduce the functions of the form f () = a( + h) n + k and to sketch graphs of this form through
Chapter 3 Eponential and Logarithmic Functions 3. Eponential Functions and Their Graphs 3.2 Logarithmic Functions and Their Graphs 3.3 Properties of Logarithms 3.4 Solving Eponential and Logarithmic Equations
Chapter 5 Polnomial and Rational Functions Section 5.1 Polnomial Functions Section summaries The general form of a polnomial function is f() = a n n + a n 1 n 1 + +a 1 + a 0. The degree of f() is the largest
1 MRE GRAPHS AND EQUATINS HI-RES STILL T BE SUPPLIED Different-shaped curves are seen in man areas of mathematics, science, engineering and the social sciences. For eample, Galileo showed that if an object
6 CHAPTER Analsis of Graphs of Functions. STRETCHING, SHRINKING, AND REFLECTING GRAPHS Vertical Stretching Vertical Shrinking Reflecting Across an Ais Combining Transformations of Graphs In the previous
- Chapter A Chapter A - Rectangular Coordinate Sstem Introduction: Rectangular Coordinate Sstem Although the use of rectangular coordinates in such geometric applications as surveing and planning has been
Section 4.5 Directional derivatives and gradient vectors (3/3/08) Overview: The partial derivatives f ( 0, 0 ) and f ( 0, 0 ) are the rates of change of z = f(,) at ( 0, 0 ) in the positive - and -directions.
Functions and Graphs CHAPTER 2 INTRODUCTION The function concept is one of the most important ideas in mathematics. The stud 2-1 Functions 2-2 Elementar Functions: Graphs and Transformations 2-3 Quadratic
APPENDIX A. Radicals and Rational Eponents 779 Appendies Overview This section contains a review of some basic algebraic skills. (You should read Section P. before reading this appendi.) Radical and rational
College Algera - MAT 161 Page: 1 Copright 009 Killoran Quadratic Functions The graph of f./ D a C C c (where a,,c are real and a 6D 0) is called a paraola. Paraola s are Smmetric over the line that passes
9.4 Solving Quadratic Equations Completing the Square Essential Question How can ou use completing the square to solve a quadratic equation? Work with a partner. a. Write the equation modeled the algera
78 Further Topics in Functions. Inverse Functions Thinking of a function as a process like we did in Section., in this section we seek another function which might reverse that process. As in real life,
4 (4-) Chapter 4 Polynomials and Eponents P( r) 0 ( r) dollars. Which law of eponents can be used to simplify the last epression? Simplify it. P( r) 7. CD rollover. Ronnie invested P dollars in a -year
Page 1 of 19 Review of Eponential and Logarithmic Functions An eponential function is a function in the form of f ( ) = for a fied ase, where > 0 and 1. is called the ase of the eponential function. The
. Linear Functions We now begin the stud of families of functions. Our first famil, linear functions, are old friends as we shall soon see. Recall from Geometr that two distinct points in the plane determine
Hperbolic functions The hperbolic functions have similar names to the trigonmetric functions, but the are defined in terms of the eponential function In this unit we define the three main hperbolic functions,
Appendi F: Solving Equations The goal of solving equations When you are trying to solve an equation like: = 4, you are trying to determine all of the numerical values of that you could plug into that equation.
. Piecewise Functions YOU WILL NEED graph paper graphing calculator GOAL Understand, interpret, and graph situations that are described b piecewise functions. LEARN ABOUT the Math A cit parking lot uses
000_00.qd /5/05 : AM Page SECTION. Graphs of Equations. GRAPHS OF EQUATIONS Sketch graphs of equations b hand. Find the - and -intercepts of graphs of equations. Write the standard forms of equations of
ORDER OF OPERATIONS In the following order: 1) Work inside the grouping smbols such as parenthesis and brackets. ) Evaluate the powers. 3) Do the multiplication and/or division in order from left to right.
7.1 Simplification of Rational Epressions and Functions 7.1 OBJECTIVES 1. Simplif a rational epression 2. Identif a rational function 3. Simplif a rational function 4. Graph a rational function Our work
8 Graphs of Quadratic Epressions: The Parabola In Topic 6 we saw that the graph of a linear function such as = 2 + 1 was a straight line. The graph of a function which is not linear therefore cannot be
Rotation of Aes For a discussion of conic sections, see Appendi. In precalculus or calculus ou ma have studied conic sections with equations of the form A C D E F Here we show that the general second-degree
Rational Eponents and Radical Functions.1 nth Roots and Rational Eponents. Properties of Rational Eponents and Radicals. Graphing Radical Functions. Solving Radical Equations and Inequalities. Performing
a p p e n d i f COORDINATE PLANES, LINES, AND LINEAR FUNCTIONS RECTANGULAR COORDINATE SYSTEMS Just as points on a coordinate line can be associated with real numbers, so points in a plane can be associated
354 5 Eponential and Logarithmic Functions Most of the functions we have considered so far have been polnomial and rational functions, with a few others involving roots or powers of polnomial or rational
MathsStart (NOTE Feb 2013: This is the old version of MathsStart. New books will be created during 2013 and 2014) Topic 3 Quadratic Functions 8 = 3 2 6 8 ( 2)( 4) ( 3) 2 1 2 4 0 (3, 1) MATHS LEARNING CENTRE
Functions and equations Assessment statements. Concept of function f : f (); domain, range, image (value). Composite functions (f g); identit function. Inverse function f.. The graph of a function; its
.6 Graphs of Functions 9.6 Graphs of Functions In Section. we defined a function as a special tpe of relation; one in which each -coordinate was matched with onl one -coordinate. We spent most of our time
Eponential Functions In this chapter we will study the eponential function and its inverse the logarithmic function. These important functions are indispensable in working with problems that involve population
Contents How You Ma Use This Resource Guide ii 9 Fractional and Quadratic Equations 1 Worksheet 9.1: Similar Figures.......................... 5 Worksheet 9.: Stretch of a Spring........................
G COORDINATE PLANES, LINES, AND LINEAR FUNCTIONS RECTANGULAR COORDINATE SYSTEMS Just as points on a coordinate line can be associated with real numbers, so points in a plane can be associated with pairs
cob97_ch0_0-9.qd // 9: AM Page 0 CHAPTER CONNECTIONS More on Functions CHAPTER OUTLINE. Analzing the Graph of a Function 06. The Toolbo Functions and Transformations 0. Absolute Value Functions, Equations,
Section P.9 Notes Page P.9 Linear Inequalities and Absolute Value Inequalities Sometimes the answer to certain math problems is not just a single answer. Sometimes a range of answers might be the answer.
00 College Board. All rights reserved. 00 College Board. All rights reserved. SUGGESTED LEARNING STRATEGIES: Shared Reading, Marking the Tet, Visualization, Interactive Word Wall Roller coasters are scar
8.7 Sstems of Non-Linear Equations and Inequalities 67 8.7 Sstems of Non-Linear Equations and Inequalities In this section, we stud sstems of non-linear equations and inequalities. Unlike the sstems of
Reteaching Masters To jump to a location in this book. Click a bookmark on the left. To print a part of the book. Click the Print button.. When the Print window opens, tpe in a range of pages to print.
36 Chapter 37 Infinite Series Eercise 5 Fourier Series Write seven terms of the Fourier series given the following coefficients.. a 4, a 3, a, a 3 ; b 4, b 3, b 3. a.6, a 5., a 3., a 3.4; b 7.5, b 5.3,
060_CH03_13-154.QXP 10/9/10 10:56 AM Page 13 3 Rectangular Coordinate Sstem and Graphs In This Chapter 3.1 The Rectangular Coordinate Sstem 3. Circles and Graphs 3.3 Equations of Lines 3.4 Variation Chapter
MODULE - 1 Quadratic Equations 6 QUADRATIC EQUATIONS In this lesson, you will study aout quadratic equations. You will learn to identify quadratic equations from a collection of given equations and write
0606_CH0_-78.QXP //0 :6 AM Page Unit Circle Trigonometr In This Chapter. The Circular Functions. Graphs of Sine and Cosine Functions. Graphs of Other Trigonometric Functions. Special Identities.5 Inverse
- Root Functions Graph each function. Compare to the parent graph. State the domain and range...5.. 5. 6 is multiplied b a value greater than, so the graph is a vertical stretch of. Another wa to identif
Families of Quadratics Objectives To understand the effects of a, b, and c on the graphs of parabolas of the form a 2 b c To use quadratic equations and graphs to analze the motion of projectiles To distinguish
CHAPTER C3: Functions Learning objectives After studing this chapter ou should: be familiar with the terms one-one and man-one mappings understand the terms domain and range for a mapping understand the
PROPERTIES OF ELLIPTIC CURVES AND THEIR USE IN FACTORING LARGE NUMBERS A ver important set of curves which has received considerabl attention in recent ears in connection with the factoring of large numbers
Lesson 6: Linear Functions and their Slope A linear function is represented b a line when graph, and represented in an where the variables have no whole number eponent higher than. Forms of a Linear Equation
Eponential and Logarithmic Functions 0 0. Algebra and Composition of Functions 0. Inverse Functions 0. Eponential Functions 0. Logarithmic Functions 0. Properties of Logarithms 0. The Irrational Number
MATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60 A Summar of Concepts Needed to be Successful in Mathematics The following sheets list the ke concepts which are taught in the specified math course. The sheets
4.1 The Graph of a Linear Equation 4.1 OBJECTIVES 1. Find three ordered pairs for an equation in two variables 2. Graph a line from three points 3. Graph a line b the intercept method 4. Graph a line that
DRAFT Unit of Study Exponents Grade: 8 Topic: Exponent operations and rules Length of Unit: days Focus of Learning Common Core Standards: Work with Radicals and Integer Exponents 8.EE. Know and apply the
A. THE STANDARD PARABOLA Graphing Quadratic Functions The graph of a quadratic function is called a parabola. The most basic graph is of the function =, as shown in Figure, and it is to this graph which
Conic Sections. Distance Formula and Circles. More on the Parabola. The Ellipse and Hperbola. Nonlinear Sstems of Equations in Two Variables. Nonlinear Inequalities and Sstems of Inequalities In Chapter,