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110quiz4solutions

110quiz4solutions - Math 110 Quiz 4 Sept 28 2006 Name...

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Math 110 Quiz 4 Sept. 28, 2006 Name: 1. (a) [1.5 points] Determine a function in the standard form f ( x ) = a ( x - h ) 2 + k for the parabola with vertex ( - 2 , 1) passing through the point (0 , 5). (b) [1.5 points] Rewrite your answer to part (a) in the general form f ( x ) = ax 2 + bx + c . Solutions: (a) Since ( h, k ) represents the vertex, our function has the form f ( x ) = a ( x + 2) 2 + 1. It remains to find the constant “ a ”. We use the fact that (0 , 5) lies on the graph of this function: 5 = a (0+2) 2 +1. So, 5 = 4 a +1, and so a = 1. This gives f ( x ) = ( x + 2) 2 + 1 (b) f ( x ) = ( x + 2) 2 + 1 = x 2 + 4 x + 4 + 1 = x 2 + 4 x + 5 2. [2 points] Let f ( x ) = - 2 x 2 - 8 x - 8. Rewrite this quadratic function in the standard form f ( x ) = a ( x - h ) 2 + k . Solution: This is one of those infamous “complete the square” problems: f ( x ) = - 2 x 2 - 8 x - 8 = - 2( x 2 + 4 x ) - 8 = - 2( x 2 + 4 x + 4 - 4) - 8 = - 2(( x + 2) 2 - 4) - 8 = - 2( x + 2) 2 + 8 - 8 = - 2( x + 2) 2 + 0 3. [2 points] Determine the vertex and the intercepts of the quadratic function g ( x ) = 2( x - 3)( x + 1) + 5. Solution: We get the function g ( x ) in general ( ax 2 + bx + c ) form: g ( x ) = 2( x - 3)( x + 1) + 5 = 2( x 2 - 2 x - 3) + 5 = 2 x 2 - 4 x - 6 + 5 = 2 x 2 - 4 x - 1
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