#$&* course Mth 163 The quadratic formula tells us that the graph of y = a t^2 + b t + c will have zeros at t = [ -b +- `sqrt(b^2 - 4 a c) ] / (2 a), and know where else; that is, the graph will pass through the t axis provided these values are real numbers, and will pass through the t axis note where else. The graph will have a vertex halfway between the zeros, at t = -b / (2a); this is the coordinate of the vertex even enough there are no real zeros. The y coordinate of the vertex is easily obtained by substituting this value of t into the form y = a t^2 + b t + c. The graph points corresponding to t values which are 1 unit to the right and to the left of the vertex will lie at vertical coordinates which are a units 'up' from the vertex (if a is negative then a units up is actually down).