Graphing Quadratic Equations
Quadratic equations create a type of graph called a parabola. Here is a picture of the most basic parabola, created by the equation y = x².
Notice the vertex at the point (0, 0). One way to graph a quadratic equation is to put in a bunch of values for x, get the corresponding y-values, plot the resulting points, and connect the dots. For an example, let’s graph the equation y = x² – 6x + 8 using this technique.
When x = 0, y = (0)² – 6(0) + 8 = 8.
When x = 1, y = (1)² – 6(1) + 8 = 1 – 6 + 8 = 3.
When x = 2, y = (2)² – 6(2) + 8 = 4 – 12 + 8 = 0.
When x = 3, y = (3)² – 6(3) + 8 = 9 – 18 + 8 = -1.
When x = 4, y = (4)² – 6(4) + 8 = 16 – 24 + 8 = 0.
When x = 5, y = (5)² – 6(5) + 8 = 25 – 30 + 8 = 3.
So from this, we have the points (0, 8), (1, 3), (2, 0), (3, -1), (4, 0), and (5, 3). The following graph shows these points in red, and the resulting graph of the equation in blue.
Notice the vertex at (3, -1). This takes a little while to do, so we have another method that’s a bit faster, but requires a little bit more knowledge.
To use this method, the equation must be in the standard form instead of the general form. Remember that the standard form looks like y = a(x – h)² + k. When the equation is in this form, the vertex will always be at the point (h, k). So if your equation is y = 2(x – 3)² + 4, the vertex of the parabola you get when you graph the equation will be at (3, 4). Notice the placement of the subtraction inside of the parentheses, and don’t let yourself get confused and make a mistake.
To clarify on this point, note that if the equation was y = -3(x + 2)² + 5, then the vertex would be at (-2, 5).
Once you have the vertex, the value a tells you whether the parabola is stretched vertically or shrunk vertically. If a is 2, for instance, then the parabola will be stretched vertically from the vertex by a factor of 2. If a is 0.1, then the parabola will be shrunk vertically from the vertex to 10% of its original height.