f x = 4 x 2 + 64 x + 262 The function g is defined by g x =

Question

f x = 4 x 2 + 64 x + 262 The function g is defined by g x = f x + 5 . For what value of x does g x reach its minimum?

Accepted Answer

Choice A is correct. It's given that g x = f x + 5 . Since f x = 4 x 2 + 64 x + 262 , it follows that f x + 5 = 4 x + 5 2 + 64 x + 5 + 262 . Expanding the quantity x + 5 2 in this equation yields f x + 5 = 4 x 2 + 10 x + 25 + 64 x + 5 + 262 . Distributing the 4 and the 64 yields f x + 5 = 4 x 2 + 40 x + 100 + 64 x + 320 + 262 . Combining like terms yields f x + 5 = 4 x 2 + 104 x + 682 . Therefore, g x = 4 x 2 + 104 x + 682 . For a quadratic function defined by an equation of the form g x = a x - h 2 + k , where a , h , and k are constants and a is positive, g x reaches its minimum, k , when the value of x is h . The equation g x = 4 x 2 + 104 x + 682 can be rewritten in this form by completing the square. This equation is equivalent to g x = 4 x 2 + 26 + 682 , or g x = 4 x 2 + 26 x + 169 - 169 + 682 . This equation can be rewritten as g x = 4 x + 13 2 - 169 + 682 , or g x = 4 x + 13 2 - 4 169 + 682 , which is equivalent to g x = 4 x + 13 2 + 6 . This equation is in the form g x = a x - h 2 + k , where a = 4 , h = - 13 , and k = 6 . Therefore, g x reaches its minimum when the value of x is - 13 . Choice B is incorrect. This is the value of x for which f x , rather than g x , reaches its minimum. Choice C is incorrect and may result from conceptual or calculation errors. Choice D is incorrect. This is the value of x for which f x - 5 , rather than f x + 5 , reaches its minimum.

Question f-x-4-