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Exploring Quadratic Graphs - Problem 1
As you go through your study of quadratics, you guys are going learn all kinds of things about; Discriminants, the Vertex, the Axis of Symmetry, the intercepts, the Real Solutions blah, blah, blah. You don’t have to know any of that stuff to make a graph. You can always graph anything in Math by making a table of values.
So what I’m going to do is choose some x numbers, and plug them in one at a time, to find the corresponding y values for this equation. It’s always a good idea to use some negatives and positives.
Another thing to be careful of, is what this means in terms of order operations. Remember the PEMDAS stuff that tells you have to do the exponents before you apply this multiplying by negative. So like for example, this is going to be -3 times itself which is +9, but then I have to negativize it. That’s a really common mistake I see in my Math classes.
What students think if they remember that a negative times a negative is a positive, so they want to write +9 there. But don’t forget it was +9, but we negativized it, it’s tricky. Same thing here with -2. -2 times -2 is +4, we’ve got to negativize it. +1 negativized. By the way negativize is not a real word, that’s something that I just made up. So then we see something similar here, +1 times +1 is +1, with a negative in front of it. Oh look now I’m seeing where my table is turning around. I can see my symmetry, so I already know without doing any more Math, that those are going to be -4 and -9.
Now that my table is done, I’m ready to get those dots on the graph, and I know this thing in the middle that 0, 0 is going to be what we call the Vertex. It’s the place where my y values are turning around.
So let’s get these on the graph, I’m going to start with the point (-3,-9) -3, 1, 2, 3, 4, 5, 6, 7, 8, 9 (-2,-4), (-1,-1), (0,0). This is where my graph started to turn around. I've reached my vertex. So instead of looking at my table, I don’t want have to keep going back and forth, I’m going to use symmetry. I know that this dot over here is going to be 2 away from my Axis of Symmetry, and it’s going to be the same y distance down. Now, same thing here, it’s going to be 3 away now in my same y distance down. The Axis of Symmetry cuts right through that vertex point, it goes right through 0,0.
So now I know my parabola looks like that, make sure you have arrows on the end and I’m all done. One thing you guys will learn, the thing I want to leave you with before you guys start doing your homework, is to look at the co-efficient in front of x. If you have a positive coefficient, your parabola will open up. If you have a negative coefficient, like we did here, that told me the parabola was going to open down, that minus sign. You’ll learn that later, it’s just a short cut. Some people think of it like a sad face, like negative is sad, so the parabola looks like a sad face. Maybe that’ll help you. Good luck with your homework.