L11 QUADRATIC FUNCTIONS A quadratic function is a function of the form f (x) = ax2 + bx + c where a, b and c are real numbers and a = 0. The domain of a quadratic function is the set of all real numbers. Particular cases: a = 1 and b = c = 0 a = −1 and b = c = 0 f (x) = x2 − 4x + 5 The equation of a quadratic function f (x) = ax2 + bx + c a = 0, can be written in the form f (x) = a(x − h)2 + k where b h=− and k = f (h) 2a The graph of the equation y = a(x − h)2 + k for a = 0 is a parabola that has vertex V (h, k) and a vertical axis. The parabola is oriented upward if a > 0 and is oriented down- ward if a < 0. Express in the form f (x) = a(x − h)2 + k and sketch the graph of the following: f (x) = 3x2 + 24x + 50 f (x) = −x2 + 6x + 3 Find the vertex, intercepts, domain and range. Graph the parabola f (x) = 2x2 + 12x + 10 Find the equation of the quadratic function that has the vertex V (−2, −3) and whose graph passes through the point P (−1, 0) Modeling with quadratic functions Let f (x) = a(x − h)2 + k, with a = 0 If a > 0 then f has a minimum value f (h) = k If a < 0 then f has a maximum value f (h) = k Suppose that a baseball is tossed straight up, and its height s (in feet) as a function of time t (in seconds) is given by s(t) = −16t2 + 64t + 6 where t = 0 corresponds to the time when the ball is released. When does the ball reach the maximum height? What is the maximum height of the ball? A builder has 800 feet of fence left over from a job. He wants to fence in a rectangular plot of land except for a 20-foot strip to be used as a driveway. Express A the are of the plot as a function of x For what x is the area A a maximum? What is the maximum area that he can enclose? Quadratic Inequalities Solve: x2 − 5x + 6 ≥ 0 −x2 + 3x + 4 < 0 Break-Even, Proﬁt or Loss The price p in dollars of selling x gallons of paint is given by the formula p(x) = 100 − .1x The cost of producing x-gallons of paint is given by C(x) = 10x + 8000 The revenue is R = xp and the company breaks even if the cost equal revenue. The company makes a proﬁt if the R > C. Find the break-even points. For what values of x does the company make a proﬁt?
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