Four Ways to Represent a Function: We can describe a specific function in the following four ways: * verbally (by a description in words);

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1 MA19, Activit 23: What is a Function? (Section 3.1, pp ) Date: Toda s Goal: Assignments: Perhaps the most useful mathematical idea for modeling the real world is the concept of a function. We eplore the idea of a function and then give its mathematical definition. Homework (Sec. 3.1): # 2,3,13,16,17,21,24,25,3,31,4,41,43,45,48,57,61 (pp ). Reading for net lecture: Sec. 3.2 (pp ). Functions All Around Us: In nearl ever phsical phenomenon we observe that one quantit depends on another. For instance the area of a circle is a function of its radius; height is a function of age; the number of bacteria in a culture is a function of time; temperature is a function of date; the price of a commodit is a function of the demand cost of mailing a package is a function of weight; for that commodit. Four Was to Represent a Function: We can describe a specific function in the following four was: * verball (b a description in words); * algebraicall (b an eplicit formula); * visuall (b a graph); * numericall (b a table of values). Definition of Function: A function f is a rule that assigns to each element in a set A eactl one element, called f(), in a set B. The set A is called the domain of f whereas the set B is called the codomain of f; f() is called the value of f at, or the image of under f. The range of f is the set of all possible values of f() as varies throughout the domain: range of f = {f() A}. input f Machine diagram of f f() output A a Arrow diagram of f b f f() B f(a) = f(b) range of f Eample 1: Epress the rule square, add 9, then take the square root in function notation. Evaluating a Function: The smbol that represents an arbitrar number in the domain of a function f is called an independent variable. The smbol that represents a number in the range of f is called a dependent variable. In the definition of a function the independent variable plas the role of a placeholder. For eample, the function f() = can be thought of as f( ) = To evaluate f at a number (epression), we substitute the number (epression) for the placeholder. Eample 2: If f() = evaluate the following: f(4) = f(a) + f(b) = f( 1) = f(a + b) = 57

2 { Eample 3: Evaluate the piecewise function f() = if 1 if > 1 at the indicated values: f( 4) = f( 1) = f() = f(1) = Eample 4: If f() = find: f(a) = f(a + h) f(a) h = f(a + h) = The Domain of a Function: The domain of a function is the set of all inputs for the function. The domain ma be stated eplicitl. For eample, if we write f() = then the domain is the set of all real numbers for which 2 5. If the function is given b an algebraic epression and the domain is not stated eplicitl, then b convention the domain is the set of all real numbers for which the epression is defined. Eample 5: Find the domain of each of the following functions: t 4 f() = 2 g(t) = + 1 t 2 + t 6 h(u) = 7 3u k() = 2 4 Eample 6 (Torricelli s Law): A tank holds 5 gallons of water, which drains from a leak at the bottom, causing the tank to empt in 2 minutes. The tank drains faster when it is nearl full because the pressure on the leak is greater. Torricelli s Law gives the volume of the water remaining in the tank after t minutes as ( V (t) = 5 1 t 2 (a) Find V () and V (2). ) 2 t 2. (b) What do our answers to part (a) represent? (c) Make a table of values of V (t) for t =, 5, 1, 15, 2. 58

3 MA19, Activit 24: Graphs of Functions (Section 3.2, pp ) Date: Toda s Goal: Assignments: The graph of a function is the most important wa to visualize a function. It gives a picture of the behavior or life histor of the function. We can read the value of f() from the graph as being the height of the graph above the point. Homework (Sec. 3.2): #1,5,8,13,17,2,24,25,39,44,53 (pp ). Reading for net lecture: Sec. 3.3 (pp ). f(6) f() (6,f(6)) (,f()) Graphing Functions: If f is a function with domain A, then the graph of f is the set of ordered pairs Eample 1: Sketch the graph of the function f() =. f(2) (2,f(2)) 2 6 graph of f = {(,f()) A}. In other words, the graph of f is the set of all points (,) such that = f(); that is, the graph of f is the graph of the equation = f(). Obtaining Information from the Graph of a Function: The values of a function are represented b the height of its graph above the -ais. So, we can read off the values of a function from its graph. In addition, the graph of a function helps us picture the domain and range of the function on the -ais and -ais as shown in the picture: Eample 2 (Hurdle Race): Three runners compete in a 1-meter hurdle race. The graph depicts the distance run as a function of time for each runner. Describe in words what the graph tells ou about this race. Who won the race? Did each runner finish the race? What do ou think happened to runner B? (m) 1 Range = f() Domain A B C 2 t (sec) Eample 3: The picture shows the graph of g() = From the graph, find the domain and range of g. Find g(4) and g( 2) If g() = 2, what is? 59

4 Graphing Piecewise Defined Functions: A piecewise defined function is defined b different formulas on different parts of its domain. As ou might epect, the graph of such a function consists of separate pieces. Eample 4: Sketch the graph of the piecewise function { 2 if 2 f() = 5 if > 2 Eample 5: Step functions occur frequentl in real-world eamples. The greatest integer function is one of them; sketch its graph: [] = greatest integer less than or equal to The graph of a function is a curve in the plane. But the question arises: Which curves in the -plane are graphs of functions? The Vertical Line Test: A curve in the coordinate plane is the graph of a function if and onl if no vertical line intersects the curve more than once. Graph of a function Eample 6: Determine which of the curves drawn below is the graph of a function of. Not a graph of a function Equations That Define Functions: Not ever equation involving two variables (sa and ) defines one of the variables as a function of the other (sa as a function of ). Eample 7: Which of the equations that follow define as a function of? = 4 = = 9 Some Functions and Their Graphs: Check the table on page 232 of our tetbook for a list of the graphs of the functions that we will frequentl encounter from now on in the course. The list includes: Linear functions, power functions, root functions, reciprocal functions, absolute value functions, and step functions. 6

5 MA19, Activit 25: Increasing and Decreasing Functions; (Section 3.3, pp ) Average Rate of Change Date: Toda s Goal: Assignments: Functions are often used to model changing quantities. We learn how to determine if a function is increasing or decreasing, and how to find the rate at which its values change as the variable change. Homework (Sec. 3.3): # 1, 4, 13, 15, 17, 19, 22, 31, 33 (pp ). Reading for net lecture: Sec. 3.4 (pp ). Increasing and Decreasing Functions: A function f is said to be increasing when its graph rises and decreasing when its graph falls. More precisel, we sa that: f is increasing on an interval I if f( 1 ) < f( 2 ) whenever 1 < 2 in I. f is decreasing on an interval I if f( 1 ) > f( 2 ) whenever 1 < 2 in I. B D f f C f( 2 ) f( 1 ) A f( 1 ) 1 2 f is increasing f( 2 ) 1 2 f is decreasing a b c d f is increasing on the intervals [a,b] and [c,d] f is decreasing on the interval [b,c]. Eample 1: The picture shows a rough graph of the temperature T of the water from a faucet as a function of the time t that has elapsed since the faucet was turned on. On which intervals is this function increasing? Where is it decreasing? Where is it constant? T( F) t (min) Eample 2: Graph the function f() = 2. State the intervals on which f is increasing and on which f is decreasing. 61

6 Average Rate of Change: We are all familiar with the concept of speed: If ou drive a distance of 12 miles in two hours, then our average speed, or rate of travel, is 12/2 = 6 miles per hour. In other words, the average speed is equal to the ratio of the distance traveled over the time elapsed: average speed = distance traveled time elapsed. Finding average rates of change is important in man contets. For instance, we ma be interested in knowing how quickl the air temperature is dropping as a storm approaches, or how fast revenues are increasing from the sale of a new product. Eample 3: The average rate of change of the function = f() between = a and = b is average rate of change = change in f(b) f(a) = change in b a The average rate of change is the slope of the secant line between = a and = b on the graph of f, that is, the line that passes through (a,f(a)) and (b,f(b)). f(b) f(a) a b a = f() b f(b) f(a) Consider the function h() = 3 2. Find the average rate of change of the function between = 2 and = 4. Eample 4: Consider the function g() = 2. Find the average rate of change of the function between = a and = a + h. Eample 5: The graph shows the depth of water W in a reservoir over a one-ear period, as a function of the number of das t since the beginning of the ear. (a) Determine the intervals on which the function W is increasing and on which it is decreasing. (b) What was the average rate of change of W between t = 1 and t = 2? W (ft) t (das) Eample 6: B looking at Eample 3, what can ou conclude about the Rate of Change of a linear function = m + b between an two points and 1? 62

7 MA19, Activit 26: Transformations of Functions (Section 3.4, pp ) Date: Toda s Goal: Assignments: We stud how certain transformations of a function affect its graph. This will give us a better understanding of how to graph functions. The transformations we stud are shifting, reflecting, and stretching. Homework (Sec. 3.4): # 1, 3, 4, 7, 11, 12, 14, 15, 17, 25, 35, 43, 53, 64, 68 (pp ). Reading for net lecture: Sec. 3.5 (pp ). Vertical Shifting: Suppose c >. To graph = f() + c, shift the graph of = f() UPWARD c units. To graph = f() c, shift the graph of = f() DOWNWARD c units. c = f() + c c = f() Horizontal Shifting: Suppose c >. To graph = f( c), shift the graph of = f() to the RIGHT c units. To graph = f(+c), shift the graph of = f() to the LEFT c units. = f() = f( c) = f( + c) = f() c c = f() c = f() Eample 1: Use the graph of = 2 to sketch the graphs of the following functions: = = 2 3 Eample 3: Use the graph of = 2 to sketch the graphs of the following functions: = ( 3) 2 = ( + 2) 2 1 Eample 2: Use the graph of = to sketch the graphs of the following functions: = + 3 = 2 Eample 4: Use the graph of = to sketch the graphs of the following functions: = + 3 =

8 Reflecting Graphs: To graph = f(), reflect the graph of = f() in the -ais. To graph = f( ), reflect the graph of = f() in the -ais = f() = f() Eample 5: Sketch the graphs of: = = = = = f() = = f( ) = Vertical Stretching and Shrinking: To graph = cf(): If c > 1, STRETCH the graph of = f() verticall b a factor of c. If < c < 1, SHRINK the graph of = f() verticall b a factor of c. = cf() = f() Horizontal Shrinking and Stretching: To graph = cf(): If c > 1, SHRINK the graph of = f() horizontall b a factor of 1/c. If < c < 1, STRETCH the graph of = f() horizontall b a factor of 1/c. = f(c) = f(c) = f() = cf() = f() = f() c > 1 < c < 1 c > 1 < c < 1 Eample 7: Use the graph of f() = 2 2 provided Eample 6: Sketch the graph of: = 2 2 = below to sketch the graph of f(2). 2 Even and Odd Functions: Let f be a function. f is even if f( ) = f() for all in the domain of f. f is odd if f( ) = f() for all in the domain of f. f( ) = f() = f() = = Eample 8: Determine whether the following functions are even or odd: = = EVEN Graph smmetric wrt -ais. f( ) ODD Graph smmetric wrt (,). 64

9 MA19, Activit 27: Quadratic Functions; (Section 3.5, pp ) Maima and Minima Date: Toda s Goal: Assignments: In this lecture we learn how to find the maimum and minimum values of quadratic functions. For a function that represents the profit in a business, we are interested in the maimum value; for a function that represents the amount of material to be used in a manifacturing process, we are interested in the minimum value. Homework (Sec. 3.5): #1,3,6,15,22,25,34,39,41,47,59,61 (pp ). Reading for net lecture: Sec. 3.6 (pp ). Graphing Quadratic Functions Using the Standard Form: A quadratic function is a function f of the form f() = a 2 + b + c, where a,b, and c are real numbers and a. The graph of an quadratic function is a parabola; it can be obtained from the graph of f() = 2 b the methods described in Activit 26. Indeed, b completing the square a quadratic function f() = a 2 + b + c can be epressed in the standard form f() = a( h) 2 + k. The graph of f is a parabola with verte (h,k); the parabola opens upward if a >, or downward if a <. k Verte (h, k) (Minimum) h f() = a( h) 2 + k, a > (Maimum) Verte (h, k) k h f() = a( h) 2 + k, a < Maimum and Minimum Values of Quadratic Functions: As the picture above shows: if a >, then the minimum value of f occurs at = h and this value is f(h) = k; if a <, then the maimum value of f occurs at = h and this value is f(h) = k. Eample 1: Epress the parabola = in standard form and sketch its graph. In particular, state the coordinates of its verte and its intercepts. Epressing a quadratic function in standard form helps us sketch its graph and find its maimum or minimum value. There is a formula for (h,k) that can be derived from the general quadratic function as follows: f() = a 2 + b + c = a ( 2 + ba ) + c = a ( 2 ba b a 2 ( = a + b 2a ) 2 + c b2 4a ) + c b2 4a Thus: h = b 2a k = 4ac b2 4a If a >, then the minimum value is f( b/2a). If a <, then the maimum value is f( b/2a). 65

10 Eample 2: Epress the parabola = in standard form and sketch its graph. In particular, state the coordinates of its verte and its intercepts. Observation 3: Let f() = a 2 +b+c, with a, be a quadratic function. Show that the -coordinate of the midpoint of the -intercepts of f (whenever the eist!) is the -coordinate of the verte of f. Eample 4: Find the maimum or minimum value of the function: f(t) = 1 49t 7t 2 g() = Eample 5: Find a function of the form f() = a 2 + b + c whose graph is a parabola with verte ( 1,2) and that passes through the point (4,16). Eample 6 (Path of a Ball): A ball is thrown across a plaing field. Its path is given b the equation = , where is the distance the ball has traveled horizontall, and is its height above ground level, both measured in feet. (a) What is the maimum height attained b the ball? (b) How far has it traveled horizontall when it hits the ground? Eample 7 (Pharmaceuticals): When a certain drug is taken orall, the concentration of the drug in the patient s bloodstream after t minutes is given b C(t) =.6t.2t 2, where t 24 and the concentration is measured in mg/l. When is the maimum serum concentration reached, and what is that maimum concentration? 66

11 MA19, Activit 28: Combining Functions (Section 3.6, pp ) Date: Toda s Goal: We learn how two functions f and g can be combined to form new functions. Assignments: Homework (Sec. 3.6): # 1,5,7,1,17,19,21,23,24,29,34,37,47 (pp ). Reading for net lecture: Sec. 3.7 (pp ). The Algebra of Functions: Let f and g be functions with domains A and B. We define new functions f + g, f g, fg, and f/g as follows: (f + g)() = f() + g() Domain A B (f g)() = f() g() Domain A B (fg)() = f()g() ( ) f () = f() g g() Domain A B Domain { A B g() } Note: Consider the above definition (f + g)() = f() + g(). The + on the left hand side stands for the operation of addition of functions. The + on the right hand side, however, stands for addition of the numbers f() and g(). Similar remarks hold true for the other definitions. Eample 1: Let us consider the functions f() = 2 2 and g() = 3 1. Find f + g, f g, fg, and f/g and their domains: Eample 2: Let us consider the functions f() = 9 2 and g() = 2 1. Find f + g, f g, fg, and f/g and their domains: The graph of the function f + g can be obtained from the graphs of f and g b graphical addition. This means that to obtain the value of f + g at an point we add the corresponding values of f() and g(), that is, the corresponding - coordinates. Similar statements can be made for the other operations on functions. Eample 3: Use graphical addition to sketch the graph of f +g. f g 67

12 Composition of Functions: Given an two functions f and g, we start with a number in the domain of g and find its image g(). If this number g() is in the domain of f, we can then calculate the value of f(g()). The result is a new function h() = f(g()) obtained b substituting g into f. It is called the composition (or composite) of f and g and is denoted b f g (read: f composed with g or f after g ) input g g() f f(g()) output Machine diagram of f g g f g g() f f(g()) (f g)() def = f(g()). WARNING: f g g f. Eample 4: Use f() = 3 5 and g() = 2 2 to evaluate: Arrow diagram of f g f(g()) = g(f()) = f(f(4)) = (g g)(2) = (f g)() = (g f)() = Eample 5: Let f = + 1 and g = 2 1. Find the functions f g, g f, and f f and their domains. Eample 6: Epress the function F() = in the form F() = f(g()). Eample 7: Find functions f and g so that f g = H if H() =

13 MA19, Activit 29: One-to-One Functions (Section 3.7, pp ) and Their Inverses Date: Toda s Goal: Assignments: We first define one-to-one functions, which in turn allows us to introduce the notion of inverse of a one-to-one function. These topic will be of particular importance when we stud eponential and logarithmic functions. Homework (Sec. 3.7): # 1,3,7,11,17,19,21,24,31,39,47,51,52 (pp ). Reading for net lecture: Sec. 4.1 (pp ). Definition of a One-One Function: A function with domain A is called a one-to-one function if no two elements of A have the same image, that is, f( 1 ) f( 2 ) whenever 1 2. An equivalent wa of writing the above condition is: If f( 1 ) = f( 2 ), then 1 = 2. For functions that can be graphed in the coordinate plane, there is a useful criterion to determine whether a function is one-to-one or not. A a b f f(b) f() f(a) B Horizontal Line Test: A function is one-to-one no horizontal line intersects its graph more than once. f() is not one-to-one f() is one-to-one Eample 1: Show that the function f() = 5 2 is one-to-one. Eample 2: Graph the function f() = ( 2) 2 3. The function is not one-to-one: Wh? Can ou restrict its domain so that the resulting function is one-to-one? (There is more than one correct answer.) The Inverse of a Function: A one-to one function f is precisel the function for which one can define a (unique) inverse function f 1 according to the following definition. A Definition of the Inverse of a Function: Let f be a one-to-one function with domain A and range B. Then its inverse function f 1 has domain B and range A and is defined b f 1 () = f() =, for an B. f = f() B f 1 If f takes to, then f 1 takes back to. I.e., f 1 undoes what f does. NOTE: f 1 does NOT mean 1 f. Eample 3: Suppose f() is a one-to-one function. If f(2) = 7, f(3) = 1, f(5) = 18, f 1 (2) = 6 find: f 1 (7) = f(6) = f 1 ( 1) = f(f 1 (18)) = If g() = 9 3, then g 1 (3) = 69

14 Propert of Inverse Functions: Let f() be a one-to-one function with domain A and range B. The inverse function f 1 () satisfies the following cancellation properties: f 1 (f()) = for ever A f(f 1 ()) = for ever B Conversel, an function f 1 () satisfing the above conditions is the inverse of f(). Eample 4: Show that the functions f() = 5 and g() = 1/5 are inverses of each other. Eample 5: Show that the functions f() = and g() = 5 1 are inverses of each other How to find the Inverse of a One-to-One Function: Eample 6: Find the inverse of = Write = f(). 2. Solve this equation for in terms of (if possible). 3. Interchange and. The resulting equation is = f 1 (). Eample 7: Find the inverse of = Eample 8: Find the inverse of = + 2. Graph of the Inverse Function: The principle of interchanging and to find the inverse function also gives us a method for obtaining the graph of f 1 from the graph of f. The graph of f 1 is obtained b reflecting the graph of f in the line =. The picture on the right hand side shows the graphs of: f() = + 4 and f 1 () = 2 4,. = Eample 9: Find the inverse of the function f() = Find the domain and range of f and f 1. Graph f and f 1 on the same cartesian plane. 4 f f 1 7

15 MA19, Activit 3: Polnomial Functions (Section 4.1, pp ) Date: and Their Graphs Toda s Goal: Assignments: Functions defined b polnomial epressions are called polnomial functions. The graphs of polnomials functions are beautiful, smooth curves that can increase and decrease several times. For this reason the are useful in modeling man real-world situations. Homework (Sec. 4.1): # 1,5,11,15,18,23,27,29,39,51,52 (pp ). Reading for net lecture: Sec. 4.2 (pp ). Polnomial Functions: A polnomial function of degree n is a function of the form P() = a n n + a n 1 n a 1 + a, where n is a non-negative integer and a n. The numbers a, a 1,..., a n are called the coefficients of the polnomial. The number a is the constant coefficient or constant term. The number a n, the coefficient of the highest power, is the leading coefficient, and the term a n n is the leading term. Graphs of Polnomials: The graphs of polnomials of degree and 1 are lines; the graphs of polnomials of degree 2 are parabolas. The greater the degree of the polnomial, the more complicated the graph can be. However, the graph of a polnomial function is alwas a smooth curve; that is, it has no breaks or sharp corners. End Behavior and the Leading Term: The polnomial P() = a n n + a n 1 n a 1 + a has the same end behavior as the monomial Q() = a n n, so its end behavior is determined b the degree n and the sign of the leading coefficient a n. IF = P() has odd degree and a n is positive, then + as + and as. IF = P() has odd degree and a n is negative, then + as and as +. IF = P() has even degree and a n is positive, then + as ±. IF = P() has even degree and a n is negative, then as ±. P() has odd degree P() has even degree Using Zeros to Graph Polnomials: Real Zeros of Polnomials: If P() is a polnomial and c is a real number, then the following are equivalent: 1. c is a zero of P(); 2. = c is a solution of P() = ; 3. c is a factor of P(); 4. = c is an -intercept of the graph of = P(). Intermediate Value Theorem for Polnomials: If P() is a polnomial function and P(a) and P(b) have opposite signs, then there eists at least one value c between a and b for which P(c) =. Eample 1: Determine the end behavior of the polnomial P() = 3( 2 4)( 1) 3. Eample 2: Use the Intermediate Value Theorem to show that the polnomial P() = has a zero in the interval [2,3]. 71

16 Guidelines for Graphing Polnomial Functions: 1. Zeros: Factor the polnomial to find all its real zeros; these are the -intercepts of the graph. Eample 3: Let P() = ( + 2)( 1)( 3). Find the zeros of P() and sketch its graph. 2. Test Points: Make a table of values for the polnomial. Include test points to determine whether the graph of the polnomial lies above or below the -ais on the intervals determined b the zeros. Include the -intercept in the table. 3. End Behavior: Determine the end behavior of the polnomial. Eample 4: Let P() = Find the zeros of P() and sketch its graph. 4. Graph: Plot the intercepts and the other points ou found in the table. Sketch a smooth curve that passes through these points and ehibits the required end behavior. Shape of the Graph Near a Zero: If the factor ( c) appears m times in the complete factorization of P(), i.e., P() = ( c) m Q() with Q(c), then c is said to be a zero of multiplicit m. IF c is a zero of even multiplicit, then the graph of = P() touches the -ais at (c,). IF c is a zero of odd multiplicit, then the graph of = P() crosses the -ais at (c,). Eample 5: Let f() = 3 ( 2 + 1)( 3) 4. List each real zero and its multiplicit. Determine if the graph crosses or touches the -ais. 2 P() = 1 ( + 5)( 2)2 1 1 P() = 1 ( + 1)3 1 Find the degree and the end behavior of the function. Find the -intercept of = f(). Local Maima and Minima of Polnomials: If a point (a,f(a)) is the highest point on the graph of f within some viewing rectangle, then f(a) is a local maimum value of f. If a point (b,f(b)) is the lowest point on the graph of f within some viewing rectangle, then f(b) is a local minimum value of f. local ma. point (a,f(a)) local min. point (b,f(b)) b a = f() Eample 6: Sketch the famil of polnomials = 3 c 2 for c =,1,2, and 3. How does the change in the value of c affect the graph? Local Etrema of Polnomials: If P() is a polnomial of degree n, then the graph of P() has at most n 1 local etrema values. 72

17 MA19, Activit 31: Dividing Polnomials (Section 4.2, pp ) Date: Toda s Goal: Assignments: So far we have been studing polnomial functions graphicall. We now begin to stud polnomials algebraicall. Most of our work will be concerned with factoring polnomials, and to factor, we need to know how to divide polnomials. Homework (Sec. 4.2): # 1,3,5,11,13,19,22,27,31,36,43,53 (pp ). Reading for net lecture: Sec. 4.3 (pp ). 23 Given the integers 23 and 5 we can divide one b the other. We obtain: 5 = or 23 = In general, if a and b are non-zero integers, then there eist unique integers q and r such that a = q b + r and r < b, where q is the quotient and r the remainder. This is the usual long division familiar from elementar arithmetic. Eample 1: Divide 63 b 12. Long Division of Polnomials: Dividing polnomials is much like the familiar process of dividing numbers. This process is the long division algorithm for polnomials. Division Algorithm: If P() and D() are polnomials, with D(), then there eist unique polnomials Q() and R(), where R() is either or of degree strictl less than the degree of D(), such that P() = Q() D() + R() The polnomials P() and D() are called the dividend and divisor, respectivel; Q() is the quotient and R() is the remainder. Eample 2: Divide the polnomial P() = b D() = 3. Eample 3: Divide the polnomial P() = b D() = (Complete the above table and check our work!) 73

18 Snthetic Division: This is a quick method of dividing polnomials; it can be used when the divisor is of the form c, where c is a number. In snthetic division we write onl the essential part of the long division table. NOTE that in snthetic division we abbreviate the polnomial P() b writing onl its coefficients. Moreover, instead of D() = c, we simpl write c. Writing c instead of c allows us to add instead of subtract! Eample 4: Use snthetic division to find the quotient Q() and the remainder R() when: f() = is divided b g() = 4. Eample 2 (Revisited): Use snthetic division to divide the polnomial P() = b D() = We obtain Q() = and R() = 11. That is, = (2 + 5)( 3) Eample 5: Use snthetic division to find the quotient Q() and the remainder R() when: f() = is divided b g() = + 3. The Remainder and Factor Theorems: Net, we see how snthetic division can be used to evaluate polnomials easil. Remainder Theorem: If the polnomial P() is divided b c, then the remainder is the value P(c). Proof: If the divisor D() is of the form c, then the remainder MUST be a constant R. Thus Eample 6: Let P() = (a) Find the quotient and the remainder when P() is divided b + 2. (b) Use the Remainder Theorem to find P( 2). P() = Q() ( c) + R. Setting = c in the above equation gives that P(c) = Q(c) + R = R. Thus P() = Q() ( c) + P(c). From the boed equation we obtain our net theorem, which sas that the zeros of a polnomial correspond to the linear factors of the polnomial. Eample 7: Use the Factor Theorem to determine whether + 2 is a factor of f() = Factor Theorem: The number c is a zero of P() if and onl if c is a factor of P(); that is, P() = Q() ( c) for some polnomial Q(). Eample 8: Find a polnomial of degree 3 that has zeros 1, 2, and 3, and in which the coefficient of 2 is 3. 74

19 MA19, Activit 32: Real Zeros of Polnomials (Section 4.3, pp ) Date: Toda s Goal: Assignments: We stud some algebraic methods that help us find the real zeros of a polnomial, and thereb factor the polnomial. Homework (Sec. 4.3): # 1,5,7,9,11,19,23,39,41,49,57,61,94 (pp ). Reading for net lecture: Review for the 3rd Midterm. Rational Zeros of Polnomials: Consider the polnomial P() = ( + 2)(2 )( 3) = From the factored form we see that the zeros of P() are 2, 2, and 3. From the epanded form we see that the constant term 12 is obtained b multipling 2 2 ( 3). This means that the zeros of the polnomial are all factors of the constant term. This observation can be generalized as follows. Rational Zeros Theorem: If the polnomial P() = a n n + a n 1 n a 1 + a Eample 1: List all possible rational zeros of P() = has integer coefficients, then ever rational zero of P() is of the form p q, where p is a factor of the constant coefficient a and q is a factor of the leading coefficient a n. Finding the Rational Zeros of a Polnomial: 1. List Possible Zeros: List all the possible rational zeros using the Rational Zeros Theorem. 2. Divide: Use snthetic division to evaluate the polnomial at each of the candidate for rational zeros that ou found in Step 1. When the remainder is, note the quotient ou have obtained. 3. Repeat: Repeat Steps 1. and 2. for the quotient. Stop when ou reach a quotient that is quadratic or factors easil, and use the quadratic formula or factor to find the remaining zeros. Eample 2: Find the real zeros of f() = Write f() in factored form and sketch its graph. Eample 3: Find the real solutions of the equation =. 75

20 Descartes Rule of Signs: Let P() be a polnomial with real coefficients. 1. The number of positive real zeros of P() is either equal to the number of variations in sign in P() or is less than that b an even whole number. 2. The number of negative real zeros of P() is either equal to the number of variations in sign in P( ) or is less than that b an even whole number. Eample 4: Use Descartes Rule of Signs to determine how man positive and how man negative real zeros the polnomial P() can have. Then determine the possible total number of real zeros. P() = P() = The Upper and Lower Bounds Theorem: Let P() be a polnomial with real coefficients. 1. If we divide P() be b (with b > ) using snthetic division, and if the row that contains the quotient and the remainder has no negative entr, then b is an upper bound for the real zeros of P(). 2. If we divide P() be a (with a < ) using snthetic division, and if the row that contains the quotient and the remainder has entries that are alternativel non-positive and non-negative, then a is a lower bound for the real zeros of P(). NOTE: The phrase alternativel non-positive and non-negative simpl means that the signs of the numbers alternate, with considered to either positive or negative as required. Eample 5: Show that the values a = 3 and b = 1 are lower and upper bounds for the real zeros of the polnomial P() =

21 MA19, Activit 33: Review (Sections ) Date: Objective: Additional Assignments: To prepare for Midterm 3, make sure that ou can solve the tpes of problems listed in Activities 33 and 34. At least one sample problem is listed b each tpe of problem. The sample problems are found in our tetbook and are reproduced here for our convenience. Make sure that ou can solve each of the sample problems without consulting our notes or eamples in the tetbook. Review all the activities leading to Midterm 3. Review all the problems in the previous online homework sets (8+9+1). Section 3.1: What is a Function? Evaluate a function at a given value. (Number 15) Evaluate a piecewise function at a given value. (Number 23) Evaluate a functional epression. (Numbers 25 and 33) Find the domain of a function. (Numbers 39, 41, 47, 52, and 53) Problem 15: Let g() = Evaluate g(2), g( 2), g(1 2 ), g(a), g(a 1), and g( 1). Problem 23: Let f() = { if 1 if > 1. Evaluate f( 4), f( 3 2 ), f( 1), f(), and f(1). Problem 25: Let f() = Evaluate f( + 2) and f() + f(2). Problem 33: Let f() = Find f(a), f(a + h), and f(a + h) f(a), where h. h Problems 39, 41, 47, and 53: Find the domain of each function. f() = 1 3 f() = h() = 2 5 g() = f() =

22 Section 3.2: Graphs of Functions Sketch the graph of a function b making a table of values (Numbers 1 and 19) Use the graph of a function to determine information about the function. (Number 23 and 25) Sketch the graph of a piecewise defined function (Numbers 41 and 45) Use the Vertical Line Test to determine whether a curve is the graph of a function. Problems 1 and 19: Sketch the graph of each function b first making a table of values. f() = 2 f() = 2 2 Problem 23: Use the graph in our tetbook for Problem 23 on page 233 to answer the following questions. (a) Find h( 2), h(), h(2), and h(3). (b) Find the domain and range of h. Problem 25: Use the graphs in our tetbook for Problem 25 on page 233 to answer the following questions. (a) Which is larger, f() or g()? (b) Which is larger, f( 3) or g( 3)? (c) For which values of is f() = g()? Problems 41 and 45: Sketch the graph of each piecewise defined function. { if f() = + 1 if > { 2 if 1 f() = 2 if > 1 Problem 53: Look at the graphs for Problem 53 on page 234 of our tetbook. For each graph, determine whether the curve is the graph of a function of. 78

23 Section 3.3: Increasing and Decreasing Functions; Average Rate of Change Given the graph of a function, determine the intervals on which the function is increasing and on which is decreasing. (Number 3) Determine the average rate of change of a function on a given interval. (Numbers 15, 21, and 23) Interpret the data in a table. (Number 33) Problem 3: Using the graph for Problem 3 on page 244 of our tetbook, determine the intervals on which the function is increasing and on which it is decreasing. Problem 15: Using the graph for Problem 15 on page 244 of our tetbook, determine the average rate of change of the function between the indicated values of the variable. Problem 21: Determine the average rate of change of f() = between = and = 1. Problem 23: Determine the average rate of change of f() = 3 2 between = 2 and = 2 + h. Problem 33: The table gives the population in a small coastal communit for the period Figures shown are for Januar 1 in each ear. Year Population , , , , (a) What was the average rate of change of population between 1991 and 1994? (b) What was the average rate of change of population between 1995 and 1997? (c) For what period of time was the population increasing? (d) For what period of time was the population decreasing? 79

24 Section 3.4: Transformations of Functions Understand how the transformations described on pages of our tetbook affect the shape of a graph. (Numbers 5, 7, 17, 19, 25, 33, and 47 ) Determine whether a function is even, odd, or neither. (Numbers 61 and 65). Fill in the blank. If a function is even, then the graph of the function is smmetric with respect to. If a function is odd, then the graph of the function is smmetric with respect to. Problems 5 and 7: Suppose the graph of f is given. Describe how the graph of each function can be obtained from the graph of f. = 2f() = 1 2 f() = f( 4) = f( + 4) 3 4 Problem 17: See the graph for problem 17 on page 256 of our tetbook. Match each equation with its graph. Give reasons for our answers. (a) = f( 4) (b) = f() + 3 (c) = 2f( + 6) Problem 19: Use the graph of f shown on page 257 of our tetbook to sketch the graphs of the following functions. (a) = f( 2) (b) = f() 2 (c) = 2f() (d) = f() + 3 (e) = f( ) (f) = 1 2f( 1) Problem 25: Eplain how the graph of g is obtained from the graph of f. (a) f() =, g() = 2 (a) f() =, g() =

25 Problem 33: Sketch the graph of f() = ( 2) 2, not b plotting points, but b starting with the graph of a standard function and appling transformations. Problem 47: Sketch the graph of f() = , not b plotting points, but b starting with the graph of a standard function and appling transformations. Problems 61 and 65: Determine whether the function f is even, odd, or neither. If f is even or odd, use smmetr to sketch the graph. f() = 2 f() = 3 Section 3.5: Quadratic Functions; Maima and Minima Sketch the graph of a parabola. Find its verte and its intercepts (Number 13) Epress a quadratic function in standard form. Use the standard form to find the function s minimum or maimum value. (Numbers 21, 23, and 31) Given a graph of a function, find all local maimum and minimum values. (Number 45) Problem 13: Sketch the graph of = and state the coordinates of its verte and its intercepts. Problems 21 and 23: A quadratic function is given. (a) Epress the quadratic function in standard form. (b) Sketch its graph. (c) Find its maimum or minimum value. f() = f() = Problem 31: Find the maimum or minimum value of f(t) = 1 49t 7t 2. Problem 45: Find all local maimum and minimum values of the function whose graph is shown. (See the graph for Eercise 45 on page 267 of our tetbook.) 81

26 MA19, Activit 34: Review (Sections ) Date: Objective: Additional Assignments: To prepare for Midterm 3, make sure that ou can solve the tpes of problems listed in Activities 33 and 34. At least one sample problem is listed b each tpe of problem. The sample problems are found in our tetbook and are reproduced here for our convenience. Make sure that ou can solve each of the sample problems without consulting our notes or eamples in the tetbook. Review all the activities leading to Midterm 3. Review all the problems in the previous online homework sets (8+9+1). Section 3.6: Combining Functions Find the sum, difference, product, and quotient of two functions and state their domains. (Numbers 1 and 3) Understand function composition. (Numbers 17 and 21) Find the domain of a composite function. (Number 33) Problems 1 and 3: Find f + g, f g, fg, and f/g and their domains. f() = 2, g() = + 2 f() = 1 + 2, g() = 1 Problems 17 and 21: Use f() = 3 5 and g() = 2 2 to evaluate the epressions. f(g()) g(f()) (f g)() (g f)() Problem 33: Find the functions f g, g f, f f, and g g and their domains for f() = 1 and g() =

27 Section 3.7: One-to-One Functions and Their Inverses Know the definition of a one-to-one function. (See page 28 of our tetbook.) Given the graph of a function, use the horizontal line test to determine whether the function is one-to-one. (Numbers 1 and 3) Understand the definition of an inverse function. (Numbers 17 and 19) Understand the Propert of Inverse functions on page 282 of our tetbook. (Number 23) Given a one-to-one function, find its inverse function. (Numbers 33 and 37) Given the graph of a one-to-one function, sketch the graph of its inverse function. (Number 65) Problems 1 and 3: Using the graphs for Problems 1 and 3 on page 286 of our tetbook, determine whether each function is one-to-one. Problem 17: Assume f is a one-to-one function. If f(2) = 7, find f 1 (7). If f 1 (3) = 1, find f( 1). Problem 19: If f() = 5 2, find f 1 (3). (You ma assume that f is a one-to-one function.) Problem 23: Use the Propert of Inverse Functions to show that f() = 2 5 and g() = of each other. are inverses Problems 33 and 37: Find the inverse function of f. f() = f() = Problem 65: Using the graph of f for Problem 65 on page 287 of our tetbook, sketch the graph of f 1. 83

28 Section 4.1: Polnomial Functions and Their Graphs Sketch the graph of a polnomial b transforming the graph of = n, where n is the degree of the polnomial. (Numbers 3 and 5) Match a polnomial with its graph. Be able to justif our answers. (Numbers 11, 13, and 15) Use the factored form of a polnomial to sketch its graph. Show all intercepts of the graph. Make sure the graph ehibits the proper end behavior. (Numbers 21, 25, and 31) Problems 3 and 5: Sketch the graph of the function b transforming an appropriate function of the form = n. Indicate all - and -intercepts on each graph. P() = ( + 2) 3 P() = Problems 11, 13, and 15: Match the polnomial function with one of the graphs on page 322 of our tetbook. Give reasons for each choice. P() = ( 2 4) R() = T() = Problems 21 and 25: Sketch the graph of the polnomial function. Make sure our graph shows all intercepts and ehibits the proper end behavior. P() = ( 3)( + 2)(3 2) P() = 1 12 ( + 2)2 ( 3) 2 Problem 31: Factor the polnomial P() = and use the factored form to find the zeros. Then sketch the graph. 84

29 Section 4.2: Dividing Polnomials Use long division to divide one polnomial b another polnomial. (Numbers 3 and 9) Use snthetic division to divide a polnomial b a polnomial of the form c. (Number 21) Understand the Remainder Theorem. (Number 35) Understand the Factor Theorem. (Numbers 43 and 53) Problem 3: Let P() = and Q() = (a) Divide P() b Q(). (b) Epress P() in the form P() = D() Q() + R(). Problem 9: Find the quotient and remainder using long division for the epression Problem 21: Find the quotient and remainder using snthetic division for the epression Problem 35: Use snthetic division and the Remainder Theorem to evaluate P(c) if P() = and c = 7. Problem 43: Use the Factor Theorem to show that 1 is a factor of P() = Problem 53: Find a polnomial of degree 3 that has zeros 1, 2, and 3, and in which the coefficient of 2 is 3. 85

30 Section 4.3: Real Zeros of Polnomials Use the Rational Zeros Theorem (see page 333 of our tetbook) to find all rational zeros of a polnomial. (Numbers 3, 13, and 23) Use the Rational Zeros Theorem and the Quadratic Formula if necessar to find all zeros of a polnomial. (Number 43) Problem 3: List all possible rational zeros given b the Rational Zeros Theorem (but don t check to see which actuall are zeros). R() = Problems 13 and 23: Find all rational zeros of the polnomial. P() = P() = Problem 43: Find all real zeros of the polnomial P() = Use the quadratic formula if necessar, as in Eample 3(a) on page 335 of our tetbook. 86