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Unformatted text preview: MAC1105: Quiz #19 Solutions
July 26, 2010 In the topright corner of a clean sheet of paper, write your name, UFID, and section number. Please use a pen with blue or black ink. When you are nished, FOLD your paper in half lengthwise and write your name on the back. 1. Consider the quadratic function q(x) = (x  3)2  4 (a) What is the vertex? h = 3; k = 4, so the vertex is (3, 4) (b) Does the graph open upwards or downwards? Upwards. (c) Find the xintercepts, if any. 0 = (x  3)2  4 (x  3)2 = 4 x  3 = 2 x = 5, 1 (d) Find the y intercepts, if any. (0  3)2  4 = (3)2  4 = 9  4 = 5 (e) Sketch the graph. 2. Write the following quadratic functions in standard form and state the vertex (Hint: a(x  h)2 + k): (a) q(x) = x2  3x + 2
3 q(x) = (x2  3x) + 2 = (x2  3x + 9 ) + 2  9 = (x  2 )2  1 . Vertex: 4 4 4 3 1 (2, 4) 9 p(x) = (3x2  9x)  6 = 3(x2  3x)  6 = 3(x2  3x + 4 )  6  3 2 51 3 51 3(x  2 )  4 . Vertex: ( 2 , 4 ) 27 4 (b) p(x) = 3x2  9x  6
= 3. Suppose a softdrink company determines that the price of a new zero1 calorie beverage obeys the demand equation p(x) =  25 x + 100, where p is in cents. Find the price the company should set to maximize its revenue. We will complete the square to obtain an equation for the revenue, but in fact it is not necessary to nish nding the function in order to answer the question. Letting R(x) represent the revenue:
1 1 1 R(x) = p(x) x =  25 x2 + 100x =  25 (x2  2500x) ==  25 (x2  2500x + 12502 1 12502 2 2 1250 ) + 25 =  25 (x  1250) + 25 . 1 The only thing we need from this expression is the 1250, since that's the x coordinate of the vertex, i.e., the number of drinks sold to maximize the revenue. Plugging x = 1250 into the demand equation, we have:
p(1250) =  1 1250 + 100 = 50 25 So the company should set the price of a soda at 50 cents. 2 ...
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 Summer '10
 Picklesimer
 Algebra, Completing The Square, Quadratic equation, Vertex, graph open upwards

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