August 2014, Questions 29
Let f be the function represented by the graph below. (See the image above).
Let g be a function such that g(x) = -1/2 x2 + 4x + 3.
Determine which function has the larger maximum value. Justify your answer.
The vertex of f(x) is (1, 6), so the maximum value of f is 6.
To find the vertex of g(x), first let's find the Axis of Symmetry, which runs through the Vertex. The formula for that is
x = -b / 2a
x = -(4) / ((2)(-1/2)). Note:: if you put this in your calculator, you need that "extra" pair of parentheses around the denominator.
x = -4 / -1 = 4
Find the value of g(4). This will be its maximum height. g(4) = -1/2(4)2 + 4(4) + 3 = 11
Therefore, g(x) has a higher maximum value than f(x).
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