Graphing a Parabola in Standard Form

Standard form of a parabola is when we see f(x) = x^2 + 5x + 4: a quadratic term with x-squared, a linear term with x, and a constant term which can be any number. When we do this, our goal is to first factor our function by grouping with parentheses. Ours would factor into (x + 1)(x + 4). Then, set them equal to zero and solve: this gives you the x-intercepts. x + 1 =0 and x + 4 = 0. When you solve to get x by itself, you get x = -1 and x = -4. Graph these and find their midpoint on the x-axis (x = -2.5). This is the x-coordinate of the vertex of the parabola. Then, use the formula y = -b/2a to find the y-coordinate of the vertex. Ours would be y = -5(2)(1) = -2.5. Thus, our vertex is (-2.5, -2.5) and we can plot that point on the graph. These three points form the foundation of the parabola: connect them and graph.

Note that this vertex would be at the bottom, so it would be a minimum of the function: it is the lowest point in the function. If the parabola was upside-down, the vertex would be a maximum because it would be at the top of the function.