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# PRECALC ALG & TRIG MAC 1147

UF

GPA 3.8

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This 17 page Class Notes was uploaded by Marquise Graham on Saturday September 19, 2015. The Class Notes belongs to MAC 1147 at University of Florida taught by Larissa Williamson in Fall. Since its upload, it has received 7 views. For similar materials see /class/207048/mac-1147-university-of-florida in Calculus and Pre Calculus at University of Florida.

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Date Created: 09/19/15

L13 Graphing Techniques Transformations Quadratic Functions and Models Vertical and Horizontal Translations c gt 0 To c Examgle Graph the function yx7371 Vertical Compressions and Stretches l MLth a e5 4 3 2f 1 2 3 4 5 Is E o l 39 39 39 l I I l I 39I 2 Example Above is the graph of y 2 xi In the same plane graph y 2x and y x The graph of y a f x a i O is obtained from the graph of y f x by multiplying by a the y coordinate of each point that is on the graph of f Compared to the graph of y f x the graph of y a f x is stretched vertically if lal gt1 compressed vertically if 0 lt al lt l 133 Horizontal Compressions and Stretches 123447372 1234 2 3 A 5 s 77 ix 9 fltxgtx3 fZx2x3 fend The graph of y fwc a i 0 is obtained from the graph of y f x by multiplying by l the x a coordinate of each point Which is on the graph off Compared to the graph of y f x the graph of y f wc is stretched horizontally if 0 lt lal lt l compressed horizontally if Ial gt 1 Re ections about the xAxis and the yAxis The graph of y fx is a re ection of the graph of y fx across the xaxis The graph of y f x is a re ection of the graph of y fx across the yaxis Example Graph the functions y x 1 y l x Quadratic Function The quadratic function is a function of the form fx axz bxc Where a b and c are real number and a i 0 Example Use the graph of fx x2 as a reference to sketch the graphs of the following functions hx 7x2 7 2 1 2 mx72x gxx278xl7xi42l By completing the square the equation of a quadratic function fxax2bxc a720 can be written in the form fX 006402 k 4ac 2 or kfh where h b and k 2a The Graph of fx 0m2 bx c a 72 O is Vertex Axis Parabola opens up if f h k is the minimum value of f Parabola opens down if f h k is the maximum value of f Compared with the graph of f x x2 the graph of fc00c2 bxc is stretched vertically if lal gt1 compressed vertically if 0 lt a lt l 137 Example Find the vertex axis intercepts domain and range Graph the parabola fx 2x2 12x10 Example Find the equation of the quadratic function that has the indicated vertex and whose graph passes through the given point Vertex 2 3 Point 1 O 139 Example Suppose that a baseball is tossed straight up and its heights in feet as a function of time t in seconds is given by st l6t2 64t 6 with t 0 corresponding to the instant when the ball is released When does the ball reach the maximum height What is the maximum height of the ball 140 Economics Demand Eguations The price p and the quantity x sold of a certain product obey the demand equation p x18 OSxS9O a Express the revenue R as a function of x b What is the revenue if 80 units are sold c What quantity x maximizes revenue What is the maximum revenue d What price should the company charge to maximize the revenue l4l Enclosing the Most Area A builder has 800 feet of fencing left over from a job He wants to fence in a rectangular plot of land except for a 20foot strip to be used as a driveway 2039 Express the area A of the plot as a function of x What is the domain of A For what x is the area A a maximum 142 L13 Graphing Techniques Transformations Quadratic Functions and Models Vertical and Horizontal Translations c gt 0 To Example Graph the function yx 3 l Vertical Compressions and Stretches Example Above is the graph of y x15 In the same plane graph y 2x and y x The graph of y a f x 1 0 is obtained from the graph of y f x by multiplying by a the y coordinate of each point that is on the graph of f Compared to the graph of y f x the graph of y a f x is stretched vertically if lal gt 1 compressed vertically if 0 lt lal lt l Hori7ontal C 39 and Stretches Re ections about the xAxis and the yAxis The graph of y f x is a re ection of the graph of y f x across the xwcis w The graph of y f x is a re ection of the graph of y f x across the ywcis 123474472 1234 4737711234 x i x i x Exam le Graph the functions 4 4 5 5 y J l fx x3 f2x 2x fxex The graph of y f ax 1 0 is obtained from the graph of y f x by multiplying by l the x y 1 x 5 a A coordinate of each point which is on the graph of Y3 2 Compared to the graph of y f x the graph of 75 74 E 72 4 l I 2 X3 4 yfax is i 2 stretched horizontally if 0 lt lal lt1 3 compressed horizontally if lal gt1 134 135 Quadratic Function The quadratic function is a function of the form fx ax2 bxc Where a b and c are real numbers and a i 0 Example Use the graph of f x x2 as a reference to sketch the graphs of the following functions hx 7x2 7 2 By completing the square the equation of a quadratic function fx axz bxc 11 0 can be written in the form fx 01x7h2 k 2 where h2 and k 401671 211 4a or kfh gxx278xl7xi42l The Graph of fx ax2 bx c a i 0 is Vertex Axis Parabola opens up if fh k is the minimum value of f Parabola opens down ifi fh k is the maximum value of f Compared with the graph of f x x2 the graph of fx ax2 bxc is stretched vertically if lal gt 1 compressed vertically if 0 lt la lt l Example Find the vertex axis intercepts domain Example Find the equation of the quadratic function and range Graph the parabola that has the indicated vertex and Whose graph passes fx 2x2 1236 10 through the given point Vertex 72 73 Point 7 l 0 Example Suppose that a baseball is tossed straight up and its heights in feet as a function of time t in seconds is given by st l6t2 64t 6 with t 0 corresponding to the instant when the ball is released When does the ball reach the maximum height What is the maximum height of the ball Economics Demand Eguations The price p and the quantity x sold of a certain product obey the demand equation p x18 0Sx 90 a Express the revenue R as a function of x b What is the revenue if 80 units are sold c What quantity x maximizes revenue What is the maximum revenue d What price should the company charge to maximize the revenue Enclosing the Most Area A builder has 800 feet of fencing left over from a job He wants to fence in a rectangular plot of land except for a 20foot strip to be used as a driveway Express the areaA of the plot as a function of x What is the domain of A For What x is the areaA a maximum

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