THE VERTEX OF A PARABOLA 3.2 THE VERTEX OF A PARABOLA

The Vertex Form of a Quadratic Function Example 1 (a) Sketch f(x) = (x + 3)2 − 4, and indicate the vertex. (b) Estimate the coordinates of the vertex from the graph. (c) Explain how the formula for f can be used to obtain the minimum of f. Solution (a) (b) The vertex of f appears to be about at the point (−3,−4). (c) Note that (x + 3)2 is always positive or zero, so (x + 3)2 takes on its smallest value when x + 3 = 0, that is, at x = −3. At this point (x + 3)2 − 4 takes on its smallest value, f(−3) = (−3 + 3)2 − 4 = 0 − 4 = −4.

The Vertex Form of a Quadratic Function The vertex form of a quadratic function is y = a(x − h)2 + k, where a, h, k are constants, a ≠ 0. The graph of this quadratic function has vertex (h, k) and axis of symmetry x = h. To convert from vertex form to standard form, we multiply out the squared term.

m(x) = 1/3(x+3)2 + 2

Example 4 For t in seconds, the height of a object in feet is given by the formula y = f(t) = −16t2 + 32t + 16. Using algebra, find the maximum height reached by the object and the time that height is reached. second

Area = bh