Vertex form is perhaps the most powerful way of expressing a quadratic equation. It is the easiest to graph because the x and y coordinate of the parabola’s vertex are constants, as well as how compressed the graph is. Additionally, it is fairly simple to get the x-intercepts. Vertex form is as follows:
y = a(x - h)^2 + k If you dissect vertex form, you’ll see three constants: a, h, and k (the x and y are, as usual, your input and output). To properly understand it, we started with a simpler equation in the form: y= ax^2 As you change the value of a, you’ll notice that the graph compresses and expands. When a is negative, the parabola reflects over the x-axis. As it approaches 0, it gets closer and closer to becoming a straight line. As it increases, it compresses. The other two constants are h and k. h is the x-coordinate of the vertex, and k is the y-coordinate. You’ll notice that in the form above h and k are both equal to 0, so the vertex is centered at the origin. If you change the value of h, it’ll shift the parabola along the x-coordinate. Additionally, if you change the value of k it’ll shift the vertex up and down the y-axis. So, when you put them all together, you get vertex form. See the gallery on the right for graphical proof of the effects of each of the constants. |
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Standard Form:
Standard form is expressed as y = ax^2+bx+c where a, b, and c are constants (a is the same a seen in vertex form), in fact c is the y intercept, and x is the input. Standard form is the neatest way to write a quadratic, and is the intermediate between vertex and factored form. Besides that, however, you don’t get too much useful information out of it without completing the square or performing some algebra on it. Factored Form: Factored form is expressed as y = a(x - p)(x - q) where a is the familiar constant we’ve seen, along with p and q. This form gives you the x intercepts quite easily, it is always p and q, but with the opposite sign. If you have the equation y = (x - 2)(x + 3), the x intercepts would be 2 and -3 (as one of the expressions must equal 0 for the y value to be 0) On the right you can see proof that all three forms yield the same graph. |
Vertex Form to Standard Form:
Standard Form to Vertex Form:
Factored Form to Standard Form:
Standard Form to Factored Form:
One technique that our teacher showed us to help with completing the square and factoring was to create an area diagram. It’s essentially a more visual form of the algebra associated with multiplying two expressions in the form (x - p)(x - q). It can even be used for multiplying larger expressions, as seen on the right. This can be really useful when you’re starting off because it is really explicit in how the multiplication is happening, but once you get the hang of it you don’t really need them anymore. |