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Exploring Quadratic Graphs - Problem 3
Graphing an equation like this is a little tricky, because there is a whole lot of things being done to x. You need to be really careful with the order of operations, when you’re doing your substitution. So it told me to make a table, I’m going to choose some x values. Personally, I tend to make a lot of mistakes with negatives, I don’t know about you. So I’m going to start with x equals 0 and I’ll go from there.
If my x value was 0, my y number would be 0 take away 3. That’s -3² to +9 plus 1 is 10, that’s the first point in my table. Then if I try 1, 1 take away 3 is -2² is 4, plus 1 is 5. 2 take away 3 is -1, -1 times itself is +1 plus 1 again. 1, 2 I’m just moving along with my x numbers.
If I plug in x equals 3, I’ll have 0² plus 1. If I plug in 4, 4 take away 3 is 1² plus 1 is 2. I’m happy because I started to find that symmetry. This 2 showed up again, so I know without having to do any more Math, how to complete my table. 5 is going to be here, 6 is going to be here, matched up with my next consecutive x numbers 4, 5, 6.
These problems even though they involve a lot of Math here in the equation, they can be filled in pretty quickly when you use short-cuts of symmetry. The last thing I know before I start making my graph, is that I’m going to have the Vertex point 3,1. So let’s get these dots on the graph.
I’m going to start with 0,10. So 0 is my side to side number and 10 is my up and down number 0, 1, 2 , 3, 4, 5, 6, 7, 8, 9, 10 that’s my y intercept. My next point was (1,5) 1, 2, 3, 4, 5. (2, 2) then I have my Vertex 3,1. That Vertex is really important because that’s where I’m going to introduce my Axis of Symmetry and start putting dots on there without having to count any more.
This vertical line that goes right through my Vertex is not on the parabola, but it helps me graph these other dots. Like this guy is one away from the Axis of Symmetry, same height. The same thing here, equally as high but 2 away, 3 away, boom, those are the points from my table. I know and I didn’t even have to count them, then I can draw my parabola that connects it. I missed that point, but you guys get the idea.
The way this is useful is because symmetry can make these graphs go a lot more quickly. Your teacher will be impressed also, if you can describe how you created this table, not by doing it all out one by one, but by doing patterns and symmetry. They not only show up in the table, but they also show up in the graph. It will make your homework be a lot more quick.