This week in precalc 11 we’ve explored different forms of quadratic equations and what information we could gather from each equation to graph the parabola it’s representing. 

General Form 

The general form can tell us only enough about our parabola. Let’s start with “a” It notifies us of the direction the parabola would open toward if there were to be a negative in front of “a” This indicates that the parable should be opening down. However since “a” is positive likewise the parabola would be opening up. We also noticed that if there were to be a coefficient in front of it would tell us the spacing values of the parabola. Right now our spacing would just be the parent function spacing which is 1,3,5 because no coefficient is visible, similarly if we have a different coefficient in front we would have to just take out the usual spacing and multiply each value of the parent function by the number visible. Let’s say we have “2a” In this case our spacing would differ from the parent function the new values would be 2,6,10. Finally “C” is the value which is the y-intercept of the parabola.

Standard Form/ Vertex Form

In this form “a” value corresponds with the explanation about it acts the same exact way. It demonstrates the stretch or compression and the spacing of the parabola. 

Let’s move on to the value “p” inside the bracket. “p” is the value that indicates the horizontal translation. In other words, it tells us where the x coordinate would (moving left or right) be relative to our parent function. Hence, it’s very important to notice that we always graph the opposing value of what we see inside the bracket. This means when we have “-p” we automatically think of moving “p” units left, but what we actually do when we graph is that we intend to take the opposite value which means we’re not moving “p” units left, but “p” right. For example, if we have “p” being replaced with 9 which will be written , but when we go to graph it we have to move 9 units right.

Now, let’s discuss the meaning of the value “q” which is pretty straightforward. “q” is the value to represent our vertical translation of the parabola which is the y coordinate (moving up or down). Note: it’s nonidentical to horizontal translation. “q” could be graph as it’s written. For instance, if we replace “q” with “+3“ we would graph that by moving up 3 units, and if it was otherwise “-3″ would would graph that by moving 3 units down. 

—> As you’ve noticed I’ve mentioned that the x-coordinate is represented by “p” and the y-coordinate is represented by “q” It’s important to know that what is meant by that is the vertex which is the most important point of the parabola and it is the very first point you plot. When we go to graph our parabola we have to start with the vertex you find that by taking the “p” and “q” and you write such as (p,q) and go on from there. 

Factored Form

Throughout “a” value represents the same thing. But in the factored form we are given the roots/solutions (zeros values) of the quadratic equation. And they are the x-intercepts of the parabola.