<p>Read the following SAT test question and then click on a button to select your answer. </p>
<p>y= -2x^2 + bx +5</p>
<p>In the xy-plane, the graph of the equation above assumes its maximum value at x = 2. What is the value of b?</p>
<p>(A) 8
(B) 4
(C) 4
(D) 8
(E) 10</p>
<p>This is a College Board SAT question. Can someone please explain how to answer this question quickly and efficiently in an understandable way. Thanks.</p>
<p>The maximum (A must be negative) of the equation y = Ax^2 + Bx + C is at x = -B/(2A). In your example A=-2 and B=b, and the value of x at the maximum is 2 so 2=(-b)/(-4) or b=8.</p>
<p>First, a general note about quadratics. Suppose I give you an equation of the type
y = ax^2 + bx + c. The maximum occurs at x = -b/2a, as is easily shown using calculus. You can also see this using Algebra I methods, which is all that’s tested on the SAT. </p>
<p>Note that y = a(x + b/2a)^2 + (c - b^2/4a), so it’s essentially the graph of y = x^2 stretched by a factor of a, shifted horizontally, and shifted vertically. So the vertex occurs at x = -b/2a, and it’s a maximum or minimum.</p>
<p>Now you can easily see (using fogcity’s argument) that b = 8.</p>