Graphing Quadratic Equations in Standard Form

What is Standard Form? The standard (vertex) form of a quadratic equation is y = a(x - h)2 + k. The vertex of the parabola that represents this equation is (h, k), and its axis of symmetry is the line x = h. If a is positive, the parabola opens upwards. If it’s negative, the parabola opens downwards.

How do We Reach Standard Form? Say we have a parabola in its general form: y = ax2 + bx + c. To convert to standard form, we first form the a(x - h)2 term. Remember, the x-coordinate of a parabola’s vertex is equal to –b/2a. Thus, h = -b/2a. a in standard form is equal to a in general form. To find k, find the value of y when x = -b/2a using the equation in its general form. This will be equal to k.

Example Convert 2x2 + 4x – 1 to standard form. Start by finding h. –b/2a = -4/(2*2) = -1. This gives us y = 2(x + 1)2 + k. To solve for k, we set x = -1 in the original equation. 2(-1)2 + 4(-1) – 1 = 2 – 4 – 1 = -3. Thus, k = -3 and the equation is: y = 2(x+1)2 – 3

Graphing Parabolas in Standard Form Quadratic equations that are in standard form are easy to graph. Standard form makes the vertex quite easy to find. You can then draw a basic parabola, dilated by a factor of a, with its vertex at (h, k).

Example Graph the equation y = 2(x+1)2 – 3. First, note that the vertex is at (-1, -3). Next, note that the parabola is dilated by a factor of 2, and thus will appear skinnier. The graph of this equation appears on the next slide.

Graph Note that the vertex of the parabola is at (-1, -3).