In this video, we’re going to look at modeling with quadratic relations are going to look at a few ways to identify them from a few different sources. First, we’re going to look at how to identify a quadratic is going to be by what type of equation it appears to be.

So if we have a linear equation, we’re going to have X values only it says with no exponent but of course, there’s an exponent of pretend one of one. We’Ll just pretend that we just have X values and when we’re talking about quadratic equations.

We’Re talking about X, values that are going to be squared, so there’s a few examples: they’re all those have X, values that have squared terms in them. So we’ve got a couple of examples here: zeppeli example B and we look at example a it has an x squared term, so that must be a quadratic example and the second one note only has an x value, so it must be a linear example.

All right. Second way we can identify a quadratic is by the type of graphic creates. So let’s take a look at some examples here we have a linear graph, it’s just a straight line and we’ll have a quadratic graph.

It’S going to be some kind of parabola whether the parabola opens up or down so three quick examples here, clearly, B and C have straight lines in them, so they must be linear and example, a has this parabola shape, so it must be a quadratic.

So let’s look at some parts of a parabola here, so there’s a parabola now first thing we’re going to identify as wear crosses the y-axis. That’S going to be our y-intercept next thing to look at is the vertex.

This is like the very bottom point or top point of our parabola and lastly, where it crosses the x-axis. Those are going to represent our roots, and we can also have this red line that goes up and down the middle, and it’s called our axis of symmetry.

Okay, so if we have a couple of examples here, just to demonstrate that when this volume from the x squared term is positive or its greater than zero, our parabola is always going to open in an upward direction, and the opposite will be true.

When we have graphs where our a value is less than zero, it’s going to be negative or graphs are always going to open down. So what’s this is going to lead us to is a couple of things refer to as the maximum and minimum so graph? That opens up as a minimum right at the vertex and a graph that opens downward has a maximum also at the vertex, so X, values would be infinity plus or minus, and our Y value, the lowest i can go, would be minus point to in this graph And over here the highest the y can go is 1.

2, and because the graph opens, we should be able to include every x value as well. Okay, last way to identify a quadratic is going to be looking at its table of values are going to be specifically looking at something called first and second differences.

So we have a linear function. Our first differences are going to be the same and we have a quadratic or second differences are going to be the same. So let’s investigate that, so here’s a table of values.

So what you want to do is you want to investigate how the y-values are changing here. So those are going to be our Y values and if you do some quick, quick mathematics, you’ll notice that from minus 1 to 1 we’re going up by three and then from one to four we’re also going up by three you’ll notice that we get a repeating Pattern of threes, so whenever we have that repeating pattern of threes, we’ve got a linear function.

So let’s take a look at the other example here now our Y values are changing and that change for the first differences is different. Instead of having a repeating pattern, we don’t have a repeating pattern, so that would be our first differences to determine our second differences.

Well, we just repeat again what we just did. We do it again and then you’ll notice. Our second differences have a consistent pattern. Then we’ve got a quadratic function. If you did not find a repeating pattern with the second differences, then you would not have a quadratic function.

In fact, you’d have an exponential function, we’ll cover that in another video. So, just taking a final look at an example here, let’s see, do we have a linear quadratic, so we take a look at our Y values, so how are they changing here looks like they’re going by threes every time.

So if our first differences are the same, then this must be a linear function. Well, taking a look over here, we are going up by one they’re here, we’re going up by three, then we’re going up by five and lastly, we went up by seven.

So our first differences aren’t the same, but if we repeat our second differences, look like we’re going to be consistently going up by twos here, and so, if our second differences are the same, we have a quadratic function.

Thank you for watching