Quadratic Functions, Parabolas, Problem Solving

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pg097 [R] G1 5-36058 / HCG / Cannon & Elich cr 11-14-95 QC3

2.5 Quadratic Functions, Parabolas, and Problem Solving 97

58. If f͑x͒ ϭ 2x ϩ 3 and the domain D of f is given by 63. Area Given the three lines y ϭ m͑x ϩ 4͒, D ϭ ͕x ͉ x 2 ϩ x Ϫ 2 Յ 0͖,then y ϭϪm͑xϩ4͒,andxϭ2, for what value of m will (a) draw a graph of f,and thethreelinesformatrianglewithanareaof24? (b) find the range of f.(Hint: What values of y occur on 64. (a) Is the point ͑Ϫ1, 3͒ on the circle the graph?) x 2 ϩ y 2 Ϫ 4x ϩ 2y Ϫ 20 ϭ 0? 59. The length L of a metal rod is a linear function of its (b) If the answer is yes, find an equation for the line temperature T, where L is measured in centimeters and that is tangent to the circle at (Ϫ1,3);iftheanswer T in degrees Celsius. The following measurements have is no, find the distance between the x-intercept been made: L ϭ 124.91 when T ϭ 0, and L ϭ 125.11 points of the circle. when T ϭ 100. 65. (a) Show that points A͑1, ͙3͒ and B͑Ϫ͙3, 1͒ are on (a) Find a formula that gives L as a function of T. the circle x 2 ϩ y 2 ϭ 4. (b) What is the length of the rod when its temperature (b) Find the area of the shaded region in the diagram. is 20Њ Celsius? (Hint: Show that segments OA and OB are perpen- (c) To what temperature should the rod be heated to dicular to each other. make it 125.17 cm long? y 60. Store Prices The owner of a grocery store finds that, A(1, 3) on average, the store can sell 872 gallons of milk per B(Ð 3, 1) week when the price per gallon is $1.98. When the price per gallon is $1.75, sales average 1125 gallons a week. Assume that the number N of gallons sold per week is x a linear function of the price P per gallon. O (a) Find a formula that gives N as a function of P. (b) If the price per gallon is $1.64, how many gallons should the store owner expect to sell per week? (c) To sell 1400 gallons per week, what price should the store set per gallon? 66. The horizontal line y ϭ 2 divides ᭝ABC into two re- 61. Equilateral Triangle Forwhatvalue(s)ofmwill the gions. Find the area of the two regions for the triangle triangle formed by the three lines y ϭϪ2, y ϭ mxϩ4, with these vertices: and y ϭϪmx ϩ 4 be equilateral? (a) A͑0, 0͒, B͑4, 0͒, C͑0, 4͒ 62. Perimeter Find m such that the three lines y ϭ mxϩ6, (b) A͑0, 0͒, B͑4, 0͒, C͑8, 4͒ y ϭϪmx ϩ 6, and y ϭ 2formatrianglewitha 67. The vertices of ᭝ABC are A͑0, 0͒, B͑4, 0͒, C͑0, 4͒.If perimeter of 16. 0ϽkϽ4, then the horizontal line y ϭ k will divide the triangle into two regions. Draw a diagram showing these regions for a typical k.Findthevalueofkfor which the areas of the two regions are equal.

2.5 QUADRATIC FUNCTIONS, PARABOLAS, AND PROBLEM SOLVING It is only fairly recently that the importance of nonlinearities has intruded itself into the world of the working scientist. Nonlinearity is one of those strange concepts that is deﬁned by what it is not. As one physicist put it, “It is like having a zoo of nonelephants.” B. J. West

Much of this course, and calculus courses to follow, deals with other inhabitants of the “zoo” of nonlinear functions, including families of polynomial, exponential, logarithmic, and trigonometric functions. All are important, but quadratic functions are among the simplest nonlinear functions mathematicians use to model the world. pg098 [V] G2 5-36058 / HCG / Cannon & Elich cr 10-31-95 QC

98 Chapter 2 Functions

I had mathematical Definition: quadratic function curiosity very early. My A quadratic function is a function with an equation equivalent to father had in his library a 2 wonderful series of f ͑x͒ ϭ ax ϩ bx ϩ c, (1) German paperback books....OnewasEuler’s where a, b,andcare real numbers and a is not zero. Algebra. I discovered by myself how to solve For example, g͑x͒ ϭ 5 Ϫ 4x 2 and h͑x͒ ϭ ͑2x Ϫ 1͒2 Ϫ x2 are quadratic func- equations. I remember that tions, but F͑x͒ ϭ ͙x 2 ϩ x ϩ 1andG͑x͒ϭ͑xϪ3͒2 Ϫ x2 are not. (In fact, G is I did this by an incredible linear.) concentration and almost painful and not-quite Basic Transformations and Graphs of Quadratic Functions conscious effort. What I did amounted to completing The graphing techniques we introduced in Section 2.3 are collectively called basic the square in my head without paper or pencil. transformations. The graph of any linear function is a line, and we will show that Stan Ulam that graph of any quadratic function can be obtained from the core parabola, f ͑x͒ ϭ x 2, by applying basic transformations. We apply terminology from the core parabola to parabolas in general. The point (0, 0) is called the vertex of the core parabola, and the y-axis is the axis of symmetry. The axis of symmetry is a help in making a hand-sketch of a parabola. Whenever we locate a point of the parabola on one side of the axis of symmetry, we automatically have another point located symmetrically on the other side. We will derive a transformation form for a general quadratic function, an equation that identifies the vertex and axis of symmetry of the graph, but to graph any particular quadratic, you may not need all of the steps. Indeed, because a graphing calculator graphs any quadratic function, we could ask why we need the transformation form at all. Because a calculator graph is dependent on the window we choose and the pixel coordinates, we need an algebraic form from which we can read more exact information. To begin, we factor out the coefficient a from the x-terms,andthenaddand subtract the square of half the resulting x-coefficient to complete the square on x. b f͑x͒ ϭ ax2 ϩ bx ϩ c ϭ a x2 ϩ x ϩ c ͩ a ͪ b b 2 b 2 ϭ a x 2 ϩ x ϩ Ϫ ϩ c ͫͩ a 4a 2ͪ 4a 2ͬ b 2 b 2 ϭ a x ϩ ϩ c Ϫ . ͩ 2aͪ ͩ 4aͪ The final equation has the form f͑x͒ ϭ a͑x Ϫ h͒2 ϩ k (2) which we recognize as a core parabola shifted so that the vertex is at the point ͑h, k͒ and the axis of symmetry is the line x ϭ h.

Parabola Features Looking at the derivation of Equation (2), we can make some observations about the graphs of quadratic functions. pg99 [R] G1 5-36058 / HCG / Cannon & Elich cr 11-14-95 QC4

2.5 Quadratic Functions, Parabolas, and Problem Solving 99

Graphs of quadratic functions For the quadratic function f͑x͒ ϭ ax2 ϩ bx ϩ c: The graph is a parabola Ϫb Axis of with axis of symmetry x ϭ . y a symmetry 2 The parabola opens upward if a Ͼ 0, downward if a Ͻ 0. Ϫb x =Ð b To ﬁnd the coordinates of the vertex, set x ϭ .Thenthey-coordinate is 2a 2a x Ϫb given by y ϭ f . ͩ 2aͪ x-intercept points The graph of every quadratic function intersects the y-axis (where x ϭ 0), but it need not have any x-intercept points. To ﬁnd any x-intercepts, we solve the equation y-intercept Vertex f͑x͒ ϭ 0. By its nature, every quadratic function has a maximum or a minimum point (depending on whether the parabola opens down or up) that occurs at the vertex of FIGURE 26 the parabola. See Figure 26.

᭤EXAMPLE 1 Locating the vertex of a parabola The graph of the quadratic function is a parabola. Locate the vertex in two ways: (i) by writing the y b function in the form of Equation (2) and (ii) by setting x ϭϪ . Sketch the 4 2a 3 graph. 2 2 2 (a) f ͑x͒ ϭ x Ϫ 2x Ϫ 3 (b) f ͑x͒ ϭ 2x ϩ 4x Ϫ 1 1 y = x2 Solution x 1 234 Ð4Ð3 Ð2 Ð1Ð1 (a) (i) Complete the square on the x-terms. Ð2 f ͑x͒ ϭ x 2 Ϫ 2x Ϫ 3 ϭ ͑x 2 Ϫ 2x ϩ 1͒ Ϫ 1 Ϫ 3 Ð3 ϭ ͑x Ϫ 1͒2 Ϫ 4, Ð4 (1, Ð 4) (a) f(x)=x2Ð2xÐ3 From this form, the graph is the core parabola shifted 1 unit right and 4 2 =(xÐ1) Ð4 units down; the vertex is at (1, Ϫ4). 2 b Ϫ2 y (ii) From f ͑x͒ ϭ x Ϫ 2x Ϫ 3, Ϫ 2a ϭϪ2·1 ϭ1. Substituting 1 for x, f ͑1͒ ϭ 1 Ϫ 2·1Ϫ3ϭϪ4, so again the vertex is the point (1, Ϫ4). The 4 y = x2 graph is shown in Figure 27(a). 3 (b) (i) Before completing the square, we factor out 2 from the x-terms: 2 2 2 1 f ͑x͒ ϭ 2x ϩ 4x Ϫ 1 ϭ 2͑x ϩ 2x͒ Ϫ 1 1 234 x ϭ 2͑x 2 ϩ 2x ϩ 1 Ϫ 1͒ Ϫ 1 ϭ 2͑x ϩ 1͒2 Ϫ 3. Ð4 Ð3 Ð2 Ð1 Ð1 The graph is obtained by shifting the core parabola 1 unit left, stretching Ð2 byafactorof2,andtranslatingthestretchedparaboladown3units;the (Ð 1, Ð 3) Ð3 vertex is at (Ϫ1, Ϫ3). Ð4 (ii) From f ͑x͒ ϭ 2x 2 ϩ 4x Ϫ 1, Ϫ b ϭϪ4 ϭϪ1, and f ͑Ϫ1͒ ϭ 2͑Ϫ1͒2 ϩ (b) f(x)=2x2+4xÐ1 2a 4 =2(x+1)2Ð3 4͑Ϫ1͒ Ϫ 1 ϭϪ3, and the vertex is at (Ϫ1, Ϫ3). The graph appears in Figure 27(b). ᭣ FIGURE 27 ᭤EXAMPLE 2 Graph, maximum or minimum, and intercept points Find the maximum or minimum value of f and the intercept points both from a calculator graph and algebraically. (a) f ͑x͒ ϭ x 2 Ϫ 2x Ϫ 3 (b) f ͑x͒ ϭ 2x 2 ϩ 4x Ϫ 1 pg100 [V] G2 5-36058 / HCG / Cannon & Elich jn 11-22-95 MP1

100 Chapter 2 Functions

Solution (a) From Example 1(a) and the graph in Figure 27(a), the minimum value of f is the y-coordinate of the vertex, Ϫ4. For the y-intercept, f͑0͒ ϭϪ3, so the y-intercept point is (0, Ϫ3). Solving the equation x 2 Ϫ 2x Ϫ 3 ϭ 0, we have ͑x Ϫ 3͒͑x ϩ 1͒ ϭ 0, so the x-intercept points are (3, 0) and (Ϫ1,0).Whilea calculator graph in any window shows that the x-intercept points are near x ϭ 3andxϭϪ1,in trace mode, we do not see the x-intercepts exactly (where the y -coordinate equals 0) unless we have a decimal window. We can zoom in as needed, though, to get as close to (3, 0) and (Ϫ1, 0) as desired. (b) From Example 1(b) and the graph in Figure 27(b), the function has a minimum value of Ϫ3. Since f͑0͒ ϭϪ1, the y-intercept point is (0, Ϫ1). Tracing alongacalculatorgraph,weﬁndthatthex-intercepts are near 0.2 and Ϫ2.2. We can get closer approximations by zooming in, but the equation 2x 2 ϩ 4x Ϫ 1 ϭ 0 does not factor with rational numbers, so we cannot read exact coordinates of the x-intercept points on any calculator graph. We use the quadratic formula to solve the equation and ﬁnd the x-intercept points exactly: Ϫ2 ϩ ͙6 Ϫ2 Ϫ ͙6 ,0 and ,0 . ᭣ ͩ 2 ͪ ͩ 2 ͪ

Strategy: (a) Drawasepa- Quadratic Functions with Limited Domain rate diagram (Figure 28b) to show a right triangle formed According to the domain convention the domain of any quadratic function is the set by altitude CD. ᭝CDB is of all real numbers unless there is some restriction. Many applications place similar to ᭝FEB. Use ratios natural restrictions on domains, as illustrated in the next two examples. of the sides to relate h and x. ᭤EXAMPLE 3 Limited domain A rectangle is inscribed in an isosceles tri- C angle ABC,asshowninFigure28a,where͉AB͉ ϭ 6and͉AC ͉ ϭ ͉ BC ͉ ϭ 5. Let x denote the width, h the height, and K the area of the rectangle. Find an equation x for (a) h as a function of x, (b) K as a function of x. (c) Find the domain of each. 5 5 Solution h ͉ CD ͉ ͉ FE ͉ D (a) Following the strategy, the ratios and are equal. ͉ DB ͉ ϭ 3, A 6 B ͉ DB ͉ ͉ EB ͉ x ͉ FE ͉ ϭ h,and͉EB ͉ ϭ 3 Ϫ .For͉CD ͉ we can apply the Pythagorean (a) 2 theorem to ᭝CDB: ͉ CD ͉ ϭ ͙52 Ϫ 32 ϭ 4. Therefore C ͉FE ͉ ͉CD ͉ h 4 2x ϭ or ϭ or h ϭ 4 Ϫ . F ͉ EB͉ ͉DB ͉ x 3 3 x 3 Ϫ 2 2 h (b) The area K is the product of x and h: x 3Ð x 2 2 2x 2 D E B K ϭ x · h ϭ x 4 Ϫ ϭ 4x Ϫ x2. ͩ 3 ͪ 3 (b) (c) From the nature of the problem, there is no rectangle unless x isapositive FIGURE 28 number less than 6. Hence the domain of both h and K is ͕x ͉ 0 Ͻ x Ͻ 6͖. ᭣ pg101 [R] G1 5-36058 / HCG / Cannon & Elich jn 11-22-95 MP1

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᭤EXAMPLE 4 Minimizing area Find the dimensions (x and h)forthe rectangle with the maximum area that can be inscribed in the isosceles triangle ABC in Figure 28a. Strategy: Graph K and Solution ﬁnd the highest point on Follow the strategy. In Example 3, we found the area K as a quadratic function of x: the graph (the vertex). 2 K͑x͒ ϭϪ x2 ϩ4x,0ϽxϽ6. 3 Graphing K as a function of x,wegetpart of a parabola that opens down. The graph of K is shown in Figure 29. The maximum value of K occurs at the vertex of the parabola, where b Ϫ4 x ϭϪ ϭ ϭ3. 2a 2͑Ϫ2͞3͒

2x 2 When x ϭ 3, h ϭ 4 Ϫ 3 ϭ 4 Ϫ ͑3͒3 ϭ 2, and K͑3͒ ϭ 6. Therefore the inscribed rectangle with the largest area has sides of lengths 3 and 2, and area 6. ᭣

Solving Quadratic Inequalities In Section 1.5, we solved quadratic inequalities by factoring quadratic expressions. Here we look at the more general situation of solving quadratic inequalities with the use of graphs, as illustrated in the following example.

y K(x)=Ð 2 x2 +4x y 3 0

FIGURE 29 FIGURE 30

᭤EXAMPLE 5 Using a graph Find the domain of the function f͑x͒ ϭ ͙x 2 Ϫ 2x Ϫ 4. Strategy: Use a graph of Solution y ϭ x 2 Ϫ 2x Ϫ 4tosee Follow the strategy. Let y ϭ x 2 Ϫ 2x Ϫ 4anddrawagraph.Thisgivesaparabola where the y-values are non- thatopensupward,withvertexat(1,Ϫ5).Toﬁndthex-intercept points, solve the negative. equation x 2 Ϫ 2x Ϫ 4 ϭ 0 by the quadratic formula to get x ϭ 1 Ϯ ͙5. Thus the intercept points are A͑1 Ϫ ͙5, 0͒ and B͑1 ϩ ͙5, 0͒,asshowninFigure30.Usethegraphtoreadoff the solution set to the inequality that deﬁnes the domain of f: D ϭ ͕x ͉ x Յ 1 Ϫ ͙5orxՆ1ϩ͙5͖ ϭ͑Ϫϱ,1Ϫ͙5͔ʜ͓1ϩ͙5, ϱ͒. ᭣ pg102 [V] G2 5-36058 / HCG / Cannon & Elich jb 11-9-95 QC2

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Maximum and Minimum Values of a Function When we use mathematical models to answer questions about an applied problem, we frequently need to determine the maximum or minimum values of a function. The general problem can be very difﬁcult, even using the tools of calculus, but when quadratic functions are involved one can simply read off a maximum or minimum value from a graph. Deﬁnition: maximum or minimum value of a function Suppose f is a function with domain D. If there is a number k in D such that f ͑k͒ Ն f ͑x͒ for every x in D,thenf͑k͒ is the maximum value of f. If there is a number k in D such that f ͑k͒ Յ f ͑x͒ for every x in D,thenf(k) is the minimum value of f.

Strategy: Firstdrawa ᭤EXAMPLE 6 A function with limited domain For the function f͑x͒ ϭ graph, being aware of the 2 5 x Ϫ 4x ϩ 2, with domain D ϭ ͕x ͉ 0 Յ x Յ 2 ͖ ﬁnd (a) the maximum and min- given domain. From the imum values of f,and(b) the set of values where f ͑x͒ Ͼ 0. graph, read off the minimum or maximum values of y. Solution (a) The graph of f isthepartoftheparabolayϭx2 Ϫ4xϩ2ontheinterval y 5 ͓0, 2͔. A calculator graph may not allow us to read all needed points in exact form, so we ﬁrst locate the vertex and then evaluate the function at the ends of

2 the domain. f(x)=x Ð4x+2 Ϫb 4 5 Using x ϭ 2a ϭ 2 ϭ 2, we have f ͑2͒ ϭϪ2, so the point (2, Ϫ2) is the 0 Յ x Յ 5 7 (0, 2) 2 vertex. At the endpoints of the domain, f͑0͒ ϭ 2, and f (2) ϭϪ4,sothegraph has endpoints (0, 2) and (5 , Ϫ 7). x 2 4 WegetthegraphshowninFigure31.Themaximumvalueoffis 2, which 5 , Ð 7 (2 4 ) occurs at the left end of the graph, and the minimum value is Ϫ2, at the vertex (2, Ð2) of the parabola. (b) To solve the inequality f͑x͒ Ͼ 0, we need the x-intercept point between 0 and FIGURE 31 1. From the quadratic formula, f͑x͒ ϭ 0 when x ϭ 2 Ϯ ͙2. Since the func- 5 tion is only deﬁned on the interval ͓0, 2͔, f͑x͒ Ͼ 0ontheinterval ͓0, 2 Ϫ ͙2͔. ᭣

᭤EXAMPLE 7 Ranges of transformed functions For the function f͑x͒ ϭ 2 5 x Ϫ 4x ϩ 2, with domain D ϭ ͕x ͉ 0 Յ x Յ 2͖, ﬁnd the range of 3 (a) y ϭ f ͑x͒ ϩ 2 (b) y ϭϪf͑x͒ (c) y ϭ f ͑x͒. 2 Solution (a) The graph of y ϭ f ͑x͒ ϩ 2 is shifted 2 units up from the graph in Figure 31 andisshowninFigure32a.Therangeoffis the interval ͓Ϫ2, 2͔, so the range of y ϭ f ͑x͒ ϩ 2 is also shifted 2 units up, to ͓0, 4͔. (b) To get the graph of y ϭϪf͑x͒,wereﬂectthegraphoffin the x-axis,asin Figure 32b. The range is still ͓Ϫ2, 2͔. 3 (c) Stretching the graph of f verticallybyafactorof2, we get part of another 3 2 3 parabola, y ϭ 2 x Ϫ 6x ϩ 3, as shown in Figure 32c. The range of y ϭ 2 f͑x͒ is ͓Ϫ3, 3͔. ᭣ pg103 [R] G1 5-36058 / HCG / Cannon & Elich jn 11-22-95 MP1

2.5 Quadratic Functions, Parabolas, and Problem Solving 103

y y y

(0, 4) (2, 2) (0, 3) 5 , 7 (2 4) 5 1 3 , x y = f(x) (2 4 ) 2 x x (2, 0) (0, Ð 2) y =Ðf(x) 5 , Ð 21 y = f(x)+2 (2 8) (2, Ð 3)

y (a) (b) (c)

6 (0, 6) FIGURE 32 5 ᭤EXAMPLE 8 Distance from a point to a line 4 2x + y =6 3 (a) Find the minimum distance from the origin to the line L given by 2x ϩ y ϭ 6. 2 P(u, v) (b) What are the coordinates of the point Q on L that is closest to the origin? 1 (3, 0) x Solution Ð1 1 2345 First draw a diagram that will help formulate the problem (Figure 33). Since (0, 0) P͑u, v͒ is on L,then2uϩvϭ6, or v ϭ 6 Ϫ 2u. FIGURE 33 d ϭ ͙͑u Ϫ 0͒2 ϩ ͑v Ϫ 0͒2 ϭ ͙u 2 ϩ v2 ϭ ͙u 2 ϩ ͑6 Ϫ 2u͒2 z ϭ ͙5u 2 Ϫ 24u ϩ 36. z =5u2Ð24u+36 10 (a) The minimum value of d will occur when the expression under the radical is a 9 minimum. Determine the minimum of the function 8 2 7 z ϭ 5u Ϫ 24u ϩ 36, (2.4, 7.2) 6 whose graph is shown in Figure 34. The lowest point on the parabola occurs 1 where u b Ϫ24 Ð1 1 234 u ϭϪ ϭϪ ϭ 2.4 2a 10 FIGURE 34 When u ϭ 2.4, z ϭ 7.2. Therefore, the minimum value of z is 7.2, so the 4 mi. minimum distance from the origin to the line L is ͙7.2 ͑Ϸ2.68͒. Island (b) The point Q on L that is closest to the origin is given by u ϭ 2.4 and x v ϭ 6 Ϫ 2͑2.4͒ ϭ 1.2. Thus, Q is point (2.4, 1.2). ᭣

Looking Ahead to Calculus Pipeline 15 mi. Not all problems lead to quadratic functions. The applied problem in the next example requires calculus techniques to ﬁnd an exact solution. With a graphing Shore calculator, however, we can ﬁnd an excellent approximation from a graph of a cost function.

᭤EXAMPLE 9 Reading a solution from a graph A freshwater pipeline is to be built from a source on shore to an island 4 miles offshore as located in the Source diagram (Figure 35). The cost of running the pipeline along the shore is $7500 per FIGURE 35 mile, but construction offshore costs $13,500 per mile. pg104 [V] G2 5-36058 / HCG / Cannon & Elich jb 11-8-95 QC1

104 Chapter 2 Functions

(a) Express the construction cost C (in thousands of dollars) as a function of x. (b) Select an appropriate window and graph y ϭ C͑x͒ to find the approximate distance x that minimizes the cost of construction. Solution (a) If we begin the underwater construction x miles from the point nearest the island, as in Figure 35, then we have ͑15 Ϫ x͒ miles along the coast, at a cost of (7500)͑15 Ϫ x͒ dollars. The distance offshore is then ͙x 2 ϩ 42,sothe offshore construction cost is (13,500)͙x 2 ϩ 16 dollars. The total cost, in thousands of dollars, is given by (2.7, 157.4) C͑x͒ ϭ 7.5͑15 Ϫ x͒ ϩ 13.5͙x 2 ϩ 16. (b) Because of the greater cost of offshore construction, it appears that x should be less than 10, so we might try an x-range of ͓0, 10͔.Wecantryavalueforxto evaluate C,sayxϭ5:C͑5͒Ϸ161.4, just over $161,000. To safely bracket that value, let’s try a y-range of ͓140, 170͔.Wegetagraphasshownin [0, 10] by [140, 170] Figure 36. Tracing to find the low point on the graph, we get a minimum of Cost: C(x) = 7.5(15 Ð x) about 157.4 near x ϭ 2.67. Furthermore, we observe that the graph is quite flat near the low point, that the cost differs only by a few dollars for x near 2.7. We + 13.5 x 2 +16 conclude that we should build along the coast for about 12.3 miles and then FIGURE 36 head directly toward the island. The cost will be about $157,400. ᭣

EXERCISES 2.5

Check Your Understanding Develop Mastery Exercises 1–5 True or False. Give reasons. Exercises 1–12 Intercept, Vertex Find the coordinates 1. If we translate the graph of y ϭ x2 two units to the right of the intercept points and vertex algebraically, and then and one unit down, the result will be the graph of y ϭ drawagraphasacheck. x 2 Ϫ 4x ϩ 3. 1. f ͑x͒ ϭ x 2 Ϫ 3 2. f ͑x͒ ϭϪx2 ϩ3 2 2. The y-intercept point for the graph of y ϭ x ϩ x Ϫ 3 3. g͑x͒ ϭ 2͑x Ϫ 1͒2 4. g͑x͒ ϭ 2͑x ϩ 1͒2 is above the x-axis. 5. f ͑x͒ ϭ ͑x ϩ 1͒2 Ϫ 3 6. f ͑x͒ ϭ ͑x Ϫ 3͒2 ϩ 1 3. The maximum value of f͑x͒ ϭ 15 Ϫ 2x Ϫ x 2 is 12. 2 4. Thegraphsofyϭx2Ϫ5xϪ4andyϭ8ϩ 7. f ͑x͒ ϭϪx Ϫ2xϩ2 3xϪx2intersect at points in Quadrants II and IV. 8. f ͑x͒ ϭ x 2 ϩ 4x ϩ 1 5. If we translate the graph of y ϭ x2 threeunitsdownit 9. f ͑x͒ ϭ 2x 2 Ϫ 4x ϩ 2 will be the graph of y ϭ 2x 2 Ϫ 6. 10. f ͑x͒ ϭϪ2x2 ϩ8xϪ5 Exercises 6–8 Fill in the blank so that the resulting state- 1 ment is true. The number of points at which the two graphs 11. f ͑x͒ ϭ x 2 ϩ 2x 2 intersect is . 1 6. f ͑x͒ ϭ x 2 Ϫ 5x Ϫ 5, g͑x͒ ϭ 8 ϩ 3x Ϫ x 2 12. f ͑x͒ ϭϪ x2 Ϫ2xϪ1 2 7. f ͑x͒ ϭ 3 ϩ 3x Ϫ x 2, g͑x͒ ϭ 8 Ϫ x 8. f ͑x͒ ϭ 3 ϩ 3x Ϫ x 2, g͑x͒ ϭ ͉ x Ϫ 2 ͉ Ϫ 2x 13. Explore For each real number b,thegraphoff͑x͒ϭ x2 Ϫbx Ϫ 1isaparabola.Chooseseveralvaluesofb Exercises 9–10 Draw a graph of function f using a greater than or equal to 1 and in each case draw the ͓Ϫ10, 10͔ by ͓Ϫ10, 10͔ window. The number of x-intercept corresponding graph. Describe the role that b plays. points visible in this window is . Where are the x and y-intercept points located? What 9. f ͑x͒ ϭ 0.3x 2 Ϫ 4x Ϫ 1 about the vertex? 10. f ͑x͒ ϭ 3 Ϫ 3x Ϫ 0.3x 2 14. Repeat Exercise 13 for f͑x͒ ϭ x 2 ϩ bx Ϫ 1. pg105 [R] G1 5-36058 / HCG / Cannon & Elich cr 11-14-95 QC3

2.5 Quadratic Functions, Parabolas, and Problem Solving 105

15. Explore Try several values of b,positiveandnega- Exercises 39–42 Verbal to Formula Give a formula for tive, and in each case draw a graph of f ͑x͒ ϭ x 2 Ϫ a quadratic function f that satisfies the specified conditions. b͉ x ͉ ϩ 4. What do you observe about the zeros of f ? The answer is not unique. 16. Explore For f ͑x͒ ϭ x 2 Ϫ 4x ϩ c, choose several 39. Both zeros of f are positive and f͑0͒ ϭϪ2. values of c and in each case draw the corresponding 40. The graph of f does not cross the x-axis and f͑0͒ ϭϪ3. graph. Describe the role that c plays. How many x-inter- 41. A zero of f is between Ϫ2andϪ1, and the other zero cept points are there? is 3. Exercises 17–20 Find the equation (in slope-intercept 42. A zero of f isgreaterthan1,theotherzeroislessthan form) for the line containing the vertex and the y-intercept Ϫ1, and the graph contains the point (0, 2). point of the graph of f. Draw a graph of f and the line as a check. Exercises 43–46 Verbal to Formula Determine the 17. f ͑x͒ ϭ x 2 Ϫ 4x ϩ 1 18. f ͑x͒ ϭ x 2 Ϫ 3x quadratic function whose graph satisfies the given conditions. 19. f ͑x͒ ϭϪ2x2 Ϫ8xϩ3 43. x ϭ Ϫ 20. f ͑x͒ ϭϪ3x2 Ϫ12x Ϫ 8 The axis of symmetry is 2,thepoint( 1, 0) is on thegraph,and(0,5)isthey-intercept point. (Hint: Use Exercises 21–24 Graphs and Quadrants Draw a graph symmetry to find the other x-intercept point, and then and determine the quadrants through which the graph of the express f ͑x͒ in factored form.) function passes. 44. Thevertexis(3,Ϫ4) and one of the x-intercept points 21. y ϭ x 2 Ϫ 4x ϩ 3 22. y ϭ 2x 2 ϩ 7x ϩ 3 is (1, 0). (See the hint in Exercise 43.) 23. y ϭ x 2 ϩ 4x ϩ 5 24. y ϭϪx2 Ϫ2xϪ1 45. The graph is obtained by translating the core parabola 3 units left and 2 units down. Exercises 25–28 Distance Between Intercepts Find the distance between the x-intercept points for the graph of the 46. The graph is obtained by reflecting the core parabola function. Solve algebraically and then check with a graph. about the x-axis,thentranslatingtotheright2units. 2 2 25. f ͑x͒ ϭ x Ϫ 4x Ϫ 3 26. f ͑x͒ ϭ x ϩ 2x Ϫ 8 Exercises 47–50 Area Let A be the y-intercept point 27. f ͑x͒ ϭ x 2 Ϫ 4x ϩ 1 and B, C be the x-intercept points for the graph of the 28. f ͑x͒ ϭϪ2x2 ϩ4xϩ3 function. Draw a diagram and then find the area of ᭝ABC. 47. f ͑x͒ ϭϪx2 Ϫxϩ6 48. f ͑x͒ ϭ x 2 Ϫ 6x ϩ 8 Exercises 29–32 Inequalities Find the solution set for 2 the given inequality. (a) Draw a graph of the left side and 49. f ͑x͒ ϭϪx Ϫ2xϩ8 read off the answer. (b) Use algebra to justify your answer. 50. f ͑x͒ ϭϪx2 Ϫ6xϩ8 29. x 2 Ϫ 4x ϩ 3 Ͼ 0 2 Exercises 51–54 Graph to Verbal and Formula Each of 30. x ϩ 5x ϩ 4 Ͻ 0 the graphs shown began with the core parabola ͑y ϭ x 2͒ 31. 2x 2 Ϫ x Ϫ 3 Ͻ 0 followed by one or more basic transformations. (a) Give a 32. Ϫx 2 ϩ 2x ϩ 4 Յ 0 verbal description of the transformation used. (b) Give an equation for the function. Check by using a graph. Exercises 33–34 Intercepts to Vertex The intercept 51. points for the graph of a quadratic function f are specified. y Find the coordinates of the vertex. 6 33. ͑Ϫ1, 0͒, ͑3, 0͒͑0, Ϫ3͒ 34. ͑Ϫ3, 0͒, ͑2, 0͒, ͑0, 6͒ 5 Exercises 35–38 Range Determine the range of the 4 function. State your answer using (a) set notation and (b) 3 interval notation. A graph will help. 2 35. f ͑x͒ ϭ x 2 ϩ 3x Ϫ 4 1 36. g͑x͒ ϭϪ2x2 ϩ4xϩ1 x Ð1 12345 37. f ͑x͒ ϭ ͑4 Ϫ x͒͑2 ϩ x͒ Ð1 Ð2 38. f ͑x͒ ϭ 2x 2 ϩ 4 ͙3 x pg106 [V] G2 5-36058 / HCG / Cannon & Elich jb 11-27-95 MP2

106 Chapter 2 Functions

52. y 61. g͑x͒ ϭ x 2 Ϫ x, Ϫ1 Յ x Յ 4 62. g͑x͒ ϭ 3x Ϫ x 2,0ϽxϽ4 2 63. f ͑x͒ ϭ 6x Ϫ x 2, x Ն 0 1 2 x 64. f ͑x͒ ϭ 12x Ϫ 3x , x Ն 0 Ð1Ð1 1 2345 Ð2 Exercises 65–66 Translations Describe horizontal and vertical translations of the graph of f so that the result will Ð3 beagraphofyϭx2.(Hint: Complete the square.) Ð4 2 2 Ð5 65. f ͑x͒ ϭ x Ϫ 4x ϩ 1 66. f ͑x͒ ϭ x ϩ 4x ϩ 5 Ð6 Exercises 67–70 Maximum, Minimum Find the maximum and/or minimum value(s) of the function. Solve alge- 53. y braically by considering the expression under the radical as 2 a quadratic function with restricted domain. Use a graph of 1 fasacheck. x 67. f ͑x͒ ϭ ͙3 ϩ 2x Ϫ x 2 Ð3 Ð2 Ð1 1 2 3 Ð1 68. f ͑x͒ ϭ ͙5 ϩ 4x Ϫ x 2 Ð2 69. g͑x͒ ϭ ͙3 ϩ 2x ϩ x 2 Ð3 70. g͑x͒ ϭ 4 Ϫ 2x ϩ x 2 Ð4 ͙ Ð5 71. Explore At the beginning of this section we noted that Ð6 F͑x͒ ϭ ͙x 2 ϩ x ϩ 1 is not a quadratic function. (a) In a decimal window, graph, in turn, each of the 54. y following: f ͑x͒ ϭ ͙x 2 ϩ x Ϫ 1 4 F͑x͒ ϭ ͙x 2 ϩ x ϩ 1 3 2 2 G͑x͒ ϭ ͙x ϩ 2x ϩ 1 1 (b) Write a paragraph to describe some of the differ- x ences between the graphs of f, F,andG.Noteany Ð1 12345 symmetries and comment on domains and on ways Ð1 Ð2 each graph differs from a parabola. Ð3 (c) The graph of G should look familiar. Explain. Ð4 72. Solve the problem in Exercise 71 where the functions are Exercises 55–58 Range, Limited Domain Aformulafor f͑x͒ ϭ ͙x2 Ϫ x Ϫ 1 a function is given along with its domain, D. Find the range F͑x͒ ϭ ͙x2 Ϫ x ϩ 1 ofthefunction.Drawagraph. G͑x͒ ϭ ͙x2 Ϫ 2x ϩ 1 55. f ͑x͒ ϭ x 2 ϩ 2x ϩ 5; 56. f ͑x͒ ϭ x 2 ϩ 2x ϩ 5; D ϭ ͕x ͉ Ϫ3 Յ x Յ 0͖ D ϭ ͕x ͉ 0 Յ x Յ 2͖ Exercises 73–74 Minimum Distance Find the minimum 57. g͑x͒ ϭϪx2 Ϫ4xϩ4; distance from point P to the graph of y ϭ x 2 ϩ 4x Ϫ 8. D ϭ ͕x ͉ Ϫ3 Ͻ x Ͻ 1͖ (Hint: Let Q͑u, v͒ be any point on the parabola. Determine a formula that gives the distance d from point P as a 58. g͑x͒ ϭϪx2 ϩ2xϩ4; D ϭ ͕x ͉ 0 Յ x Յ 3͖ function of u (do not simplify). Draw a graph and use TRACE.) 73. P͑0, 1͒ 74. P͑0, 2͒ Exercises 59–64 Maximum, Minimum Find the maxi- 75. Maximum Area Point P u, is in the ﬁrst quadrant mum and/or minimum value(s) of the function. A graph will ͑ v͒ on the graph of the line x ϩ y ϭ 4. A triangular region be helpful. is shown in the diagram. 2 59. f ͑x͒ ϭ x Ϫ 3x Ϫ 4 (a) Express the area A of the shaded region as a func- 60. g͑x͒ ϭϪx2 ϩ4xϩ3 tion of u. pg107 [R] G1 5-36058 / HCG / Cannon & Elich cr 11-13-95 QC2

2.5 Quadratic Functions, Parabolas, and Problem Solving 107

y 81. Maximum Revenue A travel agent is proposing a tour in which a group will travel in a plane of capacity 150. (0, 4) The fare will be $1400 per person if 120 or fewer people go on the tour; the fare per person for the entire group P(u, v) will be decreased by $10 for each person in excess of 120. For instance, if 125 go, the fare for each will be v (4, 0) $1400 Ϫ $10͑5͒ ϭ $1350. Let x represent the number x u of people who go on the tour and T the total revenue (in dollars) collected by the agency. Express T as a function of x.Whatvalueofxwill give a maximum total rev- (b) For what point P willtheareaoftheregionbea enue? It will be helpful to draw a graph of the function. maximum? 82. Minimum Area A piece of wire 100 centimeters long (c) What is the maximum area? is to be cut into two pieces; one of length x centimeters, 76. Repeat Exercise 75 for point P͑u, v͒ on the line segment to be formed into a circle of circumference x,andthe joining (0, 3) and (4, 0). other to be formed into a square of perimeter 100 Ϫ x centimeters. Let A represent the sum of the areas of the 77. (a) A wire 36 inches long is bent to form a rectangle. If circle and the square. x is the length of one side, ﬁnd an equation that (a) Find an equation that gives A as a function of x. gives the area A of the rectangle as a function of x. (b) Forwhatvalueofxwill A be the smallest? What is (b) Forwhatvaluesofxis the equation valid? the smallest area? 78. Maximum Area Of all the rectangles of perimeter 83. SolvetheprobleminExercise82ifthetwopiecesare 15 centimeters, ﬁnd the dimensions (length and width) to be formed into a square and an equilateral triangle. of the one with greatest area. 84. SolvetheprobleminExercise82ifthetwopiecesare 79. Maximum Area A farmer has 800 feet of fencing left to be formed into a circle and an equilateral triangle. over from an earlier job. He wants to use it to fence in a rectangular plot of land except for a 20-foot strip that 85. Maximum Capacity A long, rectangular sheet of gal- will be used for a driveway (see the diagram, where x is vanizedtin,10incheswide,istobemadeintoarain the width of the plot). Express the area A oftheplotas gutter. The two long edges will be bent at right angles a function f of x. What is the domain of f ?Forwhatx to form a rectangular trough (see diagram, which shows is A a maximum? a cross section of the gutter with height x inches).

x x x x ? 20

80. In the diagram the smaller circle is tangent to the x-axis (a) Find a formula that gives the area A of the cross at the origin and it is tangent to the larger circle, which section as a function of x. What is the domain of has a radius of 1 and center at (1, 1). What is the radius this function? of the smaller circle? (b) What value of x will give a gutter with maximum cross sectional area? Solve algebraically and use a y graphasacheck. 86. Minimum Time Aforestrangerisintheforest3miles from the nearest point P onastraightroad.Hiscaris parked down the road 5 miles from P. He can walk in theforestatarate2mi/hrandalongtheroadat5mi/hr. (a) Toward what point Q on the road between P and his (1, 1) car should he walk so that the total time T it takes to reach the car is the least? (b) How long will it take to reach the car? (Hint: Let x ͉ PQ ͉ ϭ x, then use a graph to help you solve the problem.)