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quiz 15A sol

# quiz 15A sol - Solutions to Quiz 15A...

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Solutions to Quiz 15A www.math.ufl.edu/˜harringt November 17, 2007 1. Let f ( x ) = - x 2 + x + 6 (a) Find the axis of symmetry. x = - b 2 a = - 1 2( - 1) = 1 2 Note: This is a vertical line x = 1 / 2. (b) Find the range of f . (Use interval notation.) We will first observe that the function has a local max since the leading term of f is - x 2 . Secondly, we need to find the largest value of f i.e. we need to look at the y value of the vertex. So we will consider f (1 / 2). f 1 2 = - 1 2 2 + 1 2 + 6 = - 1 4 + 2 4 + 24 4 = 25 4 Thus, the interval is ( -∞ , 25 4 ]. Notice that we include the point 25 4 in the interval. (c) Find the roots of f . Here we want to solve f ( x ) = 0. - x 2 + x + 6 = 0 x 2 - x - 6 = 0 ( x - 3)( x + 2) = 0 So x = 3 and x = - 2. 1

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2. Determine validity of the following statements. (a) The graph of f ( x ) = 4 - ( x - 2) 2 has a local max. This statement is TRUE . Consider rewriting the function as f ( x ) = - ( x - 2) 2 + 4. So the leading term is - x 2 and thus it has a local max. (b) If the discriminant b 2 - 4 ac = 0, the graph of f ( x ) = ax 2 + bx + c , ( a = 0) will touch the x -axis at its vertex.
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