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

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Unformatted text preview: Solutions to Quiz 15A www.math.ufl.edu/˜harringt November 17, 2007 1. Let f (x) = −x2 + x + 6 (a) Find the axis of symmetry. x= −1 1 −b = = 2a 2(−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 −x2 . 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). 12 1 1 2 24 25 1 =− = + +6=− + + 2 2 2 44 4 4 Thus, the interval is (−∞, 25 ]. 4 f 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. −x2 + x + 6 = 0 x2 − x − 6 = 0 (x − 3)(x + 2) = 0 So x = 3 and x = −2. 1 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 −x2 and thus it has a local max. (b) If the discriminant b2 − 4ac = 0, the graph of f (x) = ax2 + bx + c, (a = 0) will touch the x-axis at its vertex. This statement is TRUE. Let’s see why. First, we will compute the x-intercept (or commonly called the root, or zero). √ −b ± b2 − 4ac x= 2 √a −b ± 0 x= (Since the discriminant is zero) 2a −b x= 2a b So the x-int is ( −a , 0). So the vertex touches the x axis. 2 3. Find the end behavior of (x + 1)2 (x − 1)3 (3 − x)4 (2 − x)2 . We will first need to compute the leading term. The leading term is: (x)2 (x)3 (−x)4 (−x)2 = x2 x3 x4 x2 = x11 Note: The negatives cancel each other since the power of x is even. So our end behaviors look like: Disclaimer for number 3: The arrow lines are striaght and together. Please note that this is NOT correct. This is just to give an idea of where the arrows should point. This is a limitation to typing the solutions. 2 ...
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This note was uploaded on 07/12/2011 for the course MAC 1140 taught by Professor Williamson during the Fall '08 term at University of Florida.

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

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