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Math151OldSols5 - QUIZ ON CHAPTER 12 SOLUTIONS MATH 151...

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QUIZ ON CHAPTER 12 SOLUTIONS MATH 151 – SPRING 2003 – KUNIYUKI 100 POINTS TOTAL You may use mixed numbers instead of improper fractions in your answers. Don’t approximate. 1) Consider the equation x y y + - + = 6 48 93 0 2 . Its graph is a parabola in the standard xy -plane. (Show your work at the bottom of the page.) (13 points total) a) Which way does the parabola open (down, left, right, or up)? x y y x y y f y + - + = = - + - ( ) 6 48 93 0 6 48 93 2 2 1 2 44 3 44 The equation expresses x as a function of y , so the parabola either opens right or left. The leading coefficient of f y ( ) is negative a = - < ( ) 6 0 , so the parabola opens left . b) What are the coordinates of the vertex of the parabola? k y b a = = - = - - ( ) = - coordinate of vertex 2 48 2 6 4 h x f = = ( ) = - ( ) + ( ) - = - coordinate of vertex 4 6 4 48 4 93 3 2 Vertex: 3, 4 ( )
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c) What are the coordinates of the focus of the parabola? Find p , the signed distance from the vertex to the focus. p a = = - ( ) = - 1 4 1 4 6 1 24 The parabola opens left, so the focus lies to the left of the vertex. (Also, p < 0.) The vertex is at 3 4 , ( ) , so the focus lies at: 3 1 24 4 - Ê Ë Á ˆ ¯ ˜ Ê Ë Á ˆ ¯ ˜ , 2 23 24 , 4 2) Consider the equation 3 2 18 4 17 0 2 2 x y x y + + - + = . Its graph is an ellipse in the standard xy -plane. (Show your work at the bottom of the page.) (24 points total) 3 2 18 4 17 0 3 18 2 4 17 3 6 2 2 17 3 6 9 2 2 1 17 3 9 2 1 3 3 2 1 2 2 2 2 2 2 2 2 2 2 x y x y x x y y x x y y x x y y x y + + - + = + ( ) + - ( ) = - + ( ) + - ( ) = - + + ( ) + - + ( ) = - + ( ) + ( ) + ( ) + - ( ) Group terms.
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