Polynomial and Rational Functions

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1 Polnomial and Rational Functions Figure -mm film, once the standard for capturing photographic images, has been made largel obsolete b digital photograph. (credit film : modification of work b Horia Varlan; credit memor cards : modification of work b Paul Hudson) CHAPTER OUTLINE. Quadratic Functions. Power Functions and Polnomial Functions. Graphs of Polnomial Functions. Dividing Polnomials. Zeros of Polnomial Functions. Rational Functions.7 Inverses and Radical Functions.8 Modeling Using Variation Introduction Digital photograph has dramaticall changed the nature of photograph. No longer is an image etched in the emulsion on a roll of film. Instead, nearl ever aspect of recording and manipulating images is now governed b mathematics. An image becomes a series of numbers, representing the characteristics of light striking an image sensor. When we open an image file, software on a camera or computer interprets the numbers and converts them to a visual image. Photo editing software uses comple polnomials to transform images, allowing us to manipulate the image in order to crop details, change the color palette, and add special effects. Inverse functions make it possible to convert from one file format to another. In this chapter, we will learn about these concepts and discover how mathematics can be used in such applications. Download for free at

2 CHAPTER POLYNOMIAL AND RATIONAL FUNCTIONS LEARNING OBJECTIVES In this section, ou will: Recognize characteristics of parabolas. Understand how the graph of a parabola is related to its quadratic function. Determine a quadratic function s minimum or maimum value. Solve problems involving a quadratic function s minimum or maimum value.. QUADRATIC FUNCTIONS Figure An arra of satellite dishes. (credit: Matthew Colvin de Valle, Flickr) Curved antennas, such as the ones shown in Figure are commonl used to focus microwaves and radio waves to transmit television and telephone signals, as well as satellite and spacecraft communication. The cross-section of the antenna is in the shape of a parabola, which can be described b a quadratic function. In this section, we will investigate quadratic functions, which frequentl model problems involving area and projectile motion. Working with quadratic functions can be less comple than working with higher degree functions, so the provide a good opportunit for a detailed stud of function behavior. Recognizing Characteristics of Parabolas The graph of a quadratic function is a U-shaped curve called a parabola. One important feature of the graph is that it has an etreme point, called the verte. If the parabola opens up, the verte represents the lowest point on the graph, or the minimum value of the quadratic function. If the parabola opens down, the verte represents the highest point on the graph, or the maimum value. In either case, the verte is a turning point on the graph. The graph is also smmetric with a vertical line drawn through the verte, called the ais of smmetr. These features are illustrated in Figure. Ais of smmetr -intercepts intercept Verte Figure Download for free at

3 SECTION. QUADRATIC FUNCTIONS The -intercept is the point at which the parabola crosses the -ais. The -intercepts are the points at which the parabola crosses the -ais. If the eist, the -intercepts represent the zeros, or roots, of the quadratic function, the values of at which = 0. Eample Identifing the Characteristics of a Parabola Determine the verte, ais of smmetr, zeros, and -intercept of the parabola shown in Figure Figure Solution The verte is the turning point of the graph. We can see that the verte is at (, ). Because this parabola opens upward, the ais of smmetr is the vertical line that intersects the parabola at the verte. So the ais of smmetr is =. This parabola does not cross the -ais, so it has no zeros. It crosses the -ais at (0, 7) so this is the -intercept. Understanding How the Graphs of Parabolas are Related to Their Quadratic Functions The general form of a quadratic function presents the function in the form f() = a + b + c where a, b, and c are real numbers and a 0. If a > 0, the parabola opens upward. If a < 0, the parabola opens downward. We can use the general form of a parabola to find the equation for the ais of smmetr. The ais of smmetr is defined b = b b ±. If we use the quadratic formula, = _ b ac, to solve a a a + b + c = 0 for the -intercepts, or zeros, we find the value of halfwa between them is alwas = b a, the equation for the ais of smmetr. Figure represents the graph of the quadratic function written in general form as = + +. In this form, a =, b =, and c =. Because a > 0, the parabola opens upward. The ais of smmetr is = =. This () also makes sense because we can see from the graph that the vertical line = divides the graph in half. The verte alwas occurs along the ais of smmetr. For a parabola that opens upward, the verte occurs at the lowest point on the graph, in this instance, (, ). The -intercepts, those points where the parabola crosses the -ais, occur at (, 0) and (, 0). Ais of smmetr 8 = + + -intercepts Verte Figure Download for free at

4 CHAPTER POLYNOMIAL AND RATIONAL FUNCTIONS The standard form of a quadratic function presents the function in the form f() = a( h) + k where (h, k) is the verte. Because the verte appears in the standard form of the quadratic function, this form is also known as the verte form of a quadratic function. As with the general form, if a > 0, the parabola opens upward and the verte is a minimum. If a < 0, the parabola opens downward, and the verte is a maimum. Figure represents the graph of the quadratic function written in standard form as = ( + ) +. Since h = + in this eample, h =. In this form, a =, h =, and k =. Because a < 0, the parabola opens downward. The verte is at (, ). Verte = ( + ) + 8 Figure The standard form is useful for determining how the graph is transformed from the graph of =. Figure is the graph of this basic function. 0 8 = Figure If k > 0, the graph shifts upward, whereas if k < 0, the graph shifts downward. In Figure, k > 0, so the graph is shifted units upward. If h > 0, the graph shifts toward the right and if h < 0, the graph shifts to the left. In Figure, h < 0, so the graph is shifted units to the left. The magnitude of a indicates the stretch of the graph. If a >, the point associated with a particular -value shifts farther from the -ais, so the graph appears to become narrower, and there is a vertical stretch. But if a <, the point associated with a particular -value shifts closer to the -ais, so the graph appears to become wider, but in fact there is a vertical compression. In Figure, a >, so the graph becomes narrower. The standard form and the general form are equivalent methods of describing the same function. We can see this b epanding out the general form and setting it equal to the standard form. a( h) + k = a + b + c a ah + (ah + k) = a + b + c For the linear terms to be equal, the coefficients must be equal. ah = b, so h = b a. Download for free at

5 SECTION. QUADRATIC FUNCTIONS 7 This is the ais of smmetr we defined earlier. Setting the constant terms equal: ah + k = c k = c ah = c a ( a) b = c b a In practice, though, it is usuall easier to remember that k is the output value of the function when the input is h, so f(h) = k. forms of quadratic functions A quadratic function is a polnomial function of degree two. The graph of a quadratic function is a parabola. The general form of a quadratic function is f() = a + b + c where a, b, and c are real numbers and a 0. The standard form of a quadratic function is f() = a( h) + k where a 0. The verte (h, k) is located at h = b a, k = f(h) = f _ ( b a ). How To Given a graph of a quadratic function, write the equation of the function in general form.. Identif the horizontal shift of the parabola; this value is h. Identif the vertical shift of the parabola; this value is k.. Substitute the values of the horizontal and vertical shift for h and k. in the function f() = a( h) + k.. Substitute the values of an point, other than the verte, on the graph of the parabola for and f ().. Solve for the stretch factor, a.. If the parabola opens up, a > 0. If the parabola opens down, a < 0 since this means the graph was reflected about the -ais.. Epand and simplif to write in general form. Eample Writing the Equation of a Quadratic Function from the Graph Write an equation for the quadratic function g in Figure 7 as a transformation of f() =, and then epand the formula, and simplif terms to write the equation in general form. Figure 7 Solution We can see the graph of g is the graph of f() = shifted to the left and down, giving a formula in the form g() = a( ( )) = a( + ). Download for free at

6 8 CHAPTER POLYNOMIAL AND RATIONAL FUNCTIONS Substituting the coordinates of a point on the curve, such as (0, ), we can solve for the stretch factor. = a(0 + ) = a a = In standard form, the algebraic model for this graph is g () = ( + ). To write this in general polnomial form, we can epand the formula and simplif terms. g() = ( + ) = ( + )( + ) = ( + + ) = + + = + Notice that the horizontal and vertical shifts of the basic graph of the quadratic function determine the location of the verte of the parabola; the verte is unaffected b stretches and compressions. Analsis We can check our work using the table feature on a graphing utilit. First enter Y = ( + ). Net, select TBLSET, then use TblStart = and ΔTbl =, and select TABLE. See Table. 0 Table The ordered pairs in the table correspond to points on the graph. Tr It # A coordinate grid has been superimposed over the quadratic path of a basketball in Figure 8 Find an equation for the path of the ball. Does the shooter make the basket? Figure 8 (credit: modification of work b Dan Meer) How To Given a quadratic function in general form, find the verte of the parabola.. Identif a, b, and c.. Find h, the -coordinate of the verte, b substituting a and b into h = b a.. Find k, the -coordinate of the verte, b evaluating k = f (h) = f ( b a). Download for free at

7 SECTION. QUADRATIC FUNCTIONS 9 Eample Finding the Verte of a Quadratic Function Find the verte of the quadratic function f () = + 7. Rewrite the quadratic in standard form (verte form). Solution The horizontal coordinate of the verte will be at The vertical coordinate of the verte will be at h = b a = () = = k = f(h) = f ( ) = ( ) ( ) + 7 = Rewriting into standard form, the stretch factor will be the same as the a in the original quadratic. First, find the horizontal coordinate of the verte. Then find the vertical coordinate of the verte. Substitute the values into standard form, using the a from the general form. f () = a + b + c f () = + 7 The standard form of a quadratic function prior to writing the function then becomes the following: f () = ( ) + Analsis One reason we ma want to identif the verte of the parabola is that this point will inform us where the maimum or minimum value of the output occurs, k, and where it occurs,. Tr It # Given the equation g () = +, write the equation in general form and then in standard form. Finding the Domain and Range of a Quadratic Function An number can be the input value of a quadratic function. Therefore, the domain of an quadratic function is all real numbers. Because parabolas have a maimum or a minimum point, the range is restricted. Since the verte of a parabola will be either a maimum or a minimum, the range will consist of all -values greater than or equal to the -coordinate at the turning point or less than or equal to the -coordinate at the turning point, depending on whether the parabola opens up or down. domain and range of a quadratic function The domain of an quadratic function is all real numbers unless the contet of the function presents some restrictions. The range of a quadratic function written in general form f() = a + b + c with a positive a value is f () f ( a) b, or [ f ( a) b, ). The range of a quadratic function written in general form with a negative a value is f () f ( a) b, or (, f ( a) b ]. The range of a quadratic function written in standard form f () = a( h) + k with a positive a value is f () k; the range of a quadratic function written in standard form with a negative a value is f () k. Download for free at

8 0 CHAPTER POLYNOMIAL AND RATIONAL FUNCTIONS How To Given a quadratic function, find the domain and range.. Identif the domain of an quadratic function as all real numbers.. Determine whether a is positive or negative. If a is positive, the parabola has a minimum. If a is negative, the parabola has a maimum.. Determine the maimum or minimum value of the parabola, k.. If the parabola has a minimum, the range is given b f() k, or [k, ). If the parabola has a maimum, the range is given b f () k, or (, k]. Eample Finding the Domain and Range of a Quadratic Function Find the domain and range of f () = + 9. Solution As with an quadratic function, the domain is all real numbers. Because a is negative, the parabola opens downward and has a maimum value. We need to determine the maimum value. We can begin b finding the -value of the verte. The maimum value is given b f (h). The range is f () _ 0, or (, _ 0 ]. h = b a 9 = _ ( ) = 9 0 f ( 0) 9 = ( 0) 9 + 9( 0) 9 = _ 0 Tr It # Find the domain and range of f () = ( 7 ) + 8. Determining the Maimum and Minimum Values of Quadratic Functions The output of the quadratic function at the verte is the maimum or minimum value of the function, depending on the orientation of the parabola. We can see the maimum and minimum values in Figure 9. f () = ( ) + g() = ( + ) + (, ) (a) (, ) Minimum value of occurs at = Figure 9 (b) Minimum value of occurs at = There are man real-world scenarios that involve finding the maimum or minimum value of a quadratic function, such as applications involving area and revenue. Download for free at

9 SECTION. QUADRATIC FUNCTIONS Eample Finding the Maimum Value of a Quadratic Function A backard farmer wants to enclose a rectangular space for a new garden within her fenced backard. She has purchased 80 feet of wire fencing to enclose three sides, and she will use a section of the backard fence as the fourth side. a. Find a formula for the area enclosed b the fence if the sides of fencing perpendicular to the eisting fence have length L. b. What dimensions should she make her garden to maimize the enclosed area? Solution Let s use a diagram such as Figure 0 to record the given information. It is also helpful to introduce a temporar variable, W, to represent the width of the garden and the length of the fence section parallel to the backard fence. Garden W L Backard Figure 0 a. We know we have onl 80 feet of fence available, and L + W + L = 80, or more simpl, L + W = 80. This allows us to represent the width, W, in terms of L. W = 80 L Now we are read to write an equation for the area the fence encloses. We know the area of a rectangle is length multiplied b width, so A = LW = L(80 L) A(L) = 80L L This formula represents the area of the fence in terms of the variable length L. The function, written in general form, is A(L) = L + 80L. b. The quadratic has a negative leading coefficient, so the graph will open downward, and the verte will be the maimum value for the area. In finding the verte, we must be careful because the equation is not written in standard polnomial form with decreasing powers. This is wh we rewrote the function in general form above. Since a is the coefficient of the squared term, a =, b = 80, and c = 0. To find the verte: h = _ b a 80 h = _ ( ) k = A(0) = 0 and = 800 = 80(0) (0) The maimum value of the function is an area of 800 square feet, which occurs when L = 0 feet. When the shorter sides are 0 feet, there is 0 feet of fencing left for the longer side. To maimize the area, she should enclose the garden so the two shorter sides have length 0 feet and the longer side parallel to the eisting fence has length 0 feet. Analsis This problem also could be solved b graphing the quadratic function. We can see where the maimum area occurs on a graph of the quadratic function in Figure. Area (A) Download for free at (0, 800) A 0 0 Length (L) Figure 0 0

10 CHAPTER POLYNOMIAL AND RATIONAL FUNCTIONS How To Given an application involving revenue, use a quadratic equation to find the maimum.. Write a quadratic equation for a revenue function.. Find the verte of the quadratic equation.. Determine the -value of the verte. Eample Finding Maimum Revenue The unit price of an item affects its suppl and demand. That is, if the unit price goes up, the demand for the item will usuall decrease. For eample, a local newspaper currentl has 8,000 subscribers at a quarterl charge of $0. Market research has suggested that if the owners raise the price to $, the would lose,000 subscribers. Assuming that subscriptions are linearl related to the price, what price should the newspaper charge for a quarterl subscription to maimize their revenue? Solution Revenue is the amount of mone a compan brings in. In this case, the revenue can be found b multipling the price per subscription times the number of subscribers, or quantit. We can introduce variables, p for price per subscription and Q for quantit, giving us the equation Revenue = pq. Because the number of subscribers changes with the price, we need to find a relationship between the variables. We know that currentl p = 0 and Q = 8,000. We also know that if the price rises to $, the newspaper would lose,000 subscribers, giving a second pair of values, p = and Q = 79,000. From this we can find a linear equation relating the two quantities. The slope will be 79,000 8,000 m = _ 0 =,000 _ =,00 This tells us the paper will lose,00 subscribers for each dollar the raise the price. We can then solve for the -intercept. Q =,00p + b Substitute in the point Q = 8,000 and p = 0 8,000 =,00(0) + b Solve for b b = 9,000 This gives us the linear equation Q =,00p + 9,000 relating cost and subscribers. We now return to our revenue equation. Revenue = pq Revenue = p(,00p + 9,000) Revenue =,00p + 9,000p We now have a quadratic function for revenue as a function of the subscription charge. To find the price that will maimize revenue for the newspaper, we can find the verte. h = 9,000 _ (,00) =.8 The model tells us that the maimum revenue will occur if the newspaper charges $.80 for a subscription. To find what the maimum revenue is, we evaluate the revenue function. maimum revenue =,00(.8) + 9,000(.8) =,8,00 Analsis This could also be solved b graphing the quadratic as in Figure. We can see the maimum revenue on a graph of the quadratic function. Download for free at

11 SECTION. QUADRATIC FUNCTIONS Revenue ($,000),000,00,000,00, (.80,,8.) Price (p) Figure Finding the - and -Intercepts of a Quadratic Function Much as we did in the application problems above, we also need to find intercepts of quadratic equations for graphing parabolas. Recall that we find the -intercept of a quadratic b evaluating the function at an input of zero, and we find the -intercepts at locations where the output is zero. Notice in Figure that the number of -intercepts can var depending upon the location of the graph. No -intercept One -intercept Figure Number of -intercepts of a parabola Two -intercepts How To Given a quadratic function f (), find the - and -intercepts.. Evaluate f (0) to find the -intercept.. Solve the quadratic equation f () = 0 to find the -intercepts. Eample 7 Finding the - and -Intercepts of a Parabola Find the - and -intercepts of the quadratic f() = +. Solution We find the -intercept b evaluating f (0). So the -intercept is at (0, ). For the -intercepts, we find all solutions of f() = 0. f(0) = (0) + (0) = 0 = + In this case, the quadratic can be factored easil, providing the simplest method for solution. So the -intercepts are at (, 0 ) and (, 0). 0 = ( )( + ) 0 = 0 = + = or = Download for free at

12 CHAPTER POLYNOMIAL AND RATIONAL FUNCTIONS f () = Analsis B graphing the function, we can confirm that the graph crosses + (, 0) the -ais at (0, ). We can also confirm that the graph crosses the -ais, 0) at(, 0 ) and (, 0). See Figure. ) (0, ) Figure Rewriting Quadratics in Standard Form In Eample 7, the quadratic was easil solved b factoring. However, there are man quadratics that cannot be factored. We can solve these quadratics b first rewriting them in standard form. How To Given a quadratic function, find the -intercepts b rewriting in standard form.. Substitute a and b into h = b a.. Substitute = h into the general form of the quadratic function to find k.. Rewrite the quadratic in standard form using h and k.. Solve for when the output of the function will be zero to find the -intercepts. Eample 8 Finding the -Intercepts of a Parabola Find the -intercepts of the quadratic function f () = +. Solution We begin b solving for when the output will be zero. 0 = + Because the quadratic is not easil factorable in this case, we solve for the intercepts b first rewriting the quadratic in standard form. f () = a( h) + k We know that a =. Then we solve for h and k. So now we can rewrite in standard form. h = b a We can now solve for when the output will be zero. k = f ( ) = _ () = ( ) + ( ) = = f() = ( + ) 0 = ( + ) = ( + ) = ( + ) + = ± = ± The graph has -intercepts at (, 0) and ( +, 0). Download for free at

13 SECTION. QUADRATIC FUNCTIONS We can check our work b graphing the given function on a graphing utilit and observing the -intercepts. See Figure. (.7, 0) (0.7, 0) Figure Analsis We could have achieved the same results using the quadratic formula. Identif a =, b =, and c =. So the -intercepts occur at (, 0) and ( +, 0). = b ± b ac a = ± ()( ) _ () = ± 8 = ± () = ± Tr It # In a separate Tr It, we found the standard and general form for the function g() = +. Now find the - and -intercepts (if an). Eample 9 Appling the Verte and -Intercepts of a Parabola A ball is thrown upward from the top of a 0 foot high building at a speed of 80 feet per second. The ball s height above ground can be modeled b the equation H(t) = t + 80t + 0. a. When does the ball reach the maimum height? b. What is the maimum height of the ball? c. When does the ball hit the ground? Solution a. The ball reaches the maimum height at the verte of the parabola. 80 h = ( ) = 80 _ = =. The ball reaches a maimum height after. seconds. Download for free at

14 CHAPTER POLYNOMIAL AND RATIONAL FUNCTIONS b. To find the maimum height, find the -coordinate of the verte of the parabola. The ball reaches a maimum height of 0 feet. k = H ( a) b = H(.) = (.) + 80(.) + 0 = 0 c. To find when the ball hits the ground, we need to determine when the height is zero, H(t) = 0. We use the quadratic formula. t = _ 80 ± 80 ( )(0) ( ) = 80 ± 890 _ Because the square root does not simplif nicel, we can use a calculator to approimate the values of the solutions. t = _.8 or t = _ The second answer is outside the reasonable domain of our model, so we conclude the ball will hit the ground after about.8 seconds. See Figure. H 0 (., 0) H(t) = t + 80t Figure. t Tr It # Notice that the graph does not represent the phsical path of the ball upward and downward. Keep quantities on each ais in mind while interpreting the graph. A rock is thrown upward from the top of a -foot high cliff overlooking the ocean at a speed of 9 feet per second. The rock s height above ocean can be modeled b the equation H(t) = t + 9t +. a. When does the rock reach the maimum height? b. What is the maimum height of the rock? c. When does the rock hit the ocean? Access these online resources for additional instruction and practice with quadratic equations. Graphing Quadratic Functions in General Form ( Graphing Quadratic Functions in Standard Form ( Quadratic Function Review ( Characteristics of a Quadratic Function ( Download for free at

15 SECTION. SECTION EXERCISES 7. SECTION EXERCISES VERBAL. Eplain the advantage of writing a quadratic function in standard form.. Eplain wh the condition of a 0 is imposed in the definition of the quadratic function.. How can the verte of a parabola be used in solving real-world problems?. What is another name for the standard form of a quadratic function?. What two algebraic methods can be used to find the horizontal intercepts of a quadratic function? ALGEBRAIC For the following eercises, rewrite the quadratic functions in standard form and give the verte.. f () = + 7. g() = + 8. f () = 9. f () = + 0. h() = k() = 9. f () =. f () = For the following eercises, determine whether there is a minimum or maimum value to each quadratic function. Find the value and the ais of smmetr.. () = f() = 0 +. f() = f() = + 8. h(t) = t + t 9. f() = f() = + For the following eercises, determine the domain and range of the quadratic function.. f() = ( ) +. f() = ( + ). f() = + +. f() = +. k() = 9 For the following eercises, use the verte (h, k) and a point on the graph (, ) to find the general form of the equation of the quadratic function.. (h, k) = (, 0), (, ) = (, ) 7. (h, k) = (, ), (, ) = (, ) 8. (h, k) = (0, ), (, ) = (, ) 9. (h, k) = (, ), (, ) = (, ) 0. (h, k) = (, ), (, ) = (, 9). (h, k) = (, ), (, ) = (0, ). (h, k) = (0, ), (, ) = (, 0). (h, k) = (, 0), (, ) = (0, ) GRAPHICAL For the following eercises, sketch a graph of the quadratic function and give the verte, ais of smmetr, and intercepts.. f() =. f() =. f() = 7. f() = f() = f() = Download for free at

16 8 CHAPTER POLYNOMIAL AND RATIONAL FUNCTIONS For the following eercises, write the equation for the graphed function NUMERIC For the following eercises, use the table of values that represent points on the graph of a quadratic function. B determining the verte and ais of smmetr, find the general form of the equation of the quadratic function TECHNOLOGY For the following eercises, use a calculator to find the answer.. Graph on the same set of aes the functions f () =, f() =, and f() =. What appears to be the effect of changing the coefficient?. Graph on the same set of aes f() =, f() = ( ), f( ), and f() = ( + ). What appears to be the effect of adding or subtracting those numbers?. Graph on the same set of aes f() =, f() = + and f() =, f() = + and f() =. What appears to be the effect of adding a constant?. The path of an object projected at a degree angle with initial velocit of 80 feet per second is given b the function h() = (80) + where is the horizontal distance traveled and h() is the height in feet. Use the [TRACE] feature of our calculator to determine the height of the object when it has traveled 00 feet awa horizontall.. A suspension bridge can be modeled b the quadratic function h() = with where is the number of feet from the center and h() is height in feet. Use the [TRACE] feature of our calculator to estimate how far from the center does the bridge have a height of 00 feet. Download for free at

17 SECTION. SECTION EXERCISES 9 EXTENSIONS For the following eercises, use the verte of the graph of the quadratic function and the direction the graph opens to find the domain and range of the function.. Verte (, ), opens up. 7. Verte (, ) opens down. 8. Verte (, ), opens down. 9. Verte ( 00, 00), opens up. For the following eercises, write the equation of the quadratic function that contains the given point and has the same shape as the given function. 0. Contains (, ) and has shape of f() =. Verte is on the -ais.. Contains (, ) and has the shape of f() =. Verte is on the -ais.. Contains (, ) and has the shape of f() =. Verte is on the -ais. REAL-WORLD APPLICATIONS. Contains (, ) and has the shape of f() =. Verte is on the -ais.. Contains (, ) and has the shape of f() =. Verte is on the -ais.. Contains (, ) has the shape of f() =. Verte has -coordinate of.. Find the dimensions of the rectangular corral producing the greatest enclosed area given 00 feet of fencing. 8. Find the dimensions of the rectangular corral producing the greatest enclosed area split into pens of the same size given 00 feet of fencing. 70. Among all of the pairs of numbers whose difference is, find the pair with the smallest product. What is the product? 7. A rocket is launched in the air. Its height, in meters above sea level, as a function of time, in seconds, is given b h(t) =.9t + 9t +. Find the maimum height the rocket attains. 7. A soccer stadium holds,000 spectators. With a ticket price of $, the average attendance has been,000. When the price dropped to $9, the average attendance rose to,000. Assuming that attendance is linearl related to ticket price, what ticket price would maimize revenue? 7. Find the dimensions of the rectangular corral split into pens of the same size producing the greatest possible enclosed area given 00 feet of fencing. 9. Among all of the pairs of numbers whose sum is, find the pair with the largest product. What is the product? 7. Suppose that the price per unit in dollars of a cell phone production is modeled b p = $ 0.0, where is in thousands of phones produced, and the revenue represented b thousands of dollars is R = ċ p. Find the production level that will maimize revenue. 7. A ball is thrown in the air from the top of a building. Its height, in meters above ground, as a function of time, in seconds, is given b h(t) =.9t + t + 8. How long does it take to reach maimum height? 7. A farmer finds that if she plants 7 trees per acre, each tree will ield 0 bushels of fruit. She estimates that for each additional tree planted per acre, the ield of each tree will decrease b bushels. How man trees should she plant per acre to maimize her harvest? Download for free at