MATH 133 – Calculus with Fundamentals 1 Practice on Quadratic Functions and Completing the Square September 10, 2015

Background

Recall that (basic form of) the algebraic method of completing the square gives a way to rewrite any quadratic function x^{2} + bx + c as

x^{2} + bx + c = (x + r)^{2} + s

for some constants r, s. If we have a quadratic function ax^{2} + bx + c with a ≠ 1 (but a ≠ 0 as well), then we can factor out the a and apply the basic form of the process.

Questions

1) For each of the following quadratic functions f(x), complete the square, then use that form to generate a sketch of the graph y = f(x) by shifting and scaling the graph y = x^{2}.

(a) f(x) = x^{2} + 4x + 5

(b) f(x) = 2x^{2} − 5x + 2

2) A ball thrown straight upward with initial velocity 96 ft/sec has height y(t) = 96t − 16t^{2} (ft) after t (sec). What is the maximum height attained by the ball? (Hint: Complete the square in the formula for y(t) and think similarly to what you did in problem 1.)