Graph each inequality. y

Question

Graph each inequality. y < 3x^2 + 2

Accepted Answer

Hello everybody. Everything right. Today, today we're going to be looking at this question that states draw the graph or the given inequality where our inequality is Y is equal to or why is less than five X squared plus three. Now, the first thing I'll note is that we have a strict inequality. And what does that mean? Well, when you have inequalities where we have a less than sign or a greater than sign, those are called strict inequalities if it was less than, or equal to or greater than, or equal to, those are non strict inequalities. Now, for a strict and quality that go on, we have, we will have a dashed graph and I'll get more into that once we start graphing, but that's just something that we should note right away. Now, the first thing I'll do is change my lesson sign for an equal to sign just for the sake of solving the inequality. So we have Y equals five X squared plus three. And now I can identify a parent function. Now, a parent function is the function where or, or the function that our Y is equal to function is based off of. Now, in this case, I see that the inequality that was given to us, we have a plus three which is a transformation. And the five multiplying the X squared is also a transformation. So our parent function here would be Y equals X squared. And we have different rules and different things that we know about Y equals X squared. For example, we know that the graph of X squared is a parabola with a vertex at the origin. And we also know the general graph of our parent function. And I'll just draw it very quickly. So we draw, we'll have a Y axis and an X axis and we'll have a vertex at the origin and then we draw a parabola. So a Parabola looks kind of like a U where the values on the left are the same values on the right. It's mirrored, it has symmetry across the Y axis. So with this information, I can go ahead and move on now and see how we go from our parent function to our current function. So the first thing I'll add it to my parent function is the five multiplying it. So I'll have that Y is equal to five X squared now. And this means that X squared is stretched vertically by five units. And now I can finally add the plus three to get to our current function. So we'll have that Y equals five X squared plus three. That means that five X squared shifts up three units. So now we know how we go from our parent function to our current function. We know the different transformations that we gotta do now because we have an inequality, we know that we're going to have to shade our region in because there's not going to be just one answer as the answer to an inequality, there's multiple. So we have to test the point to see if that point falls within the inequality or not within the inequality. And that'll tell us where to shade our region. So I just test the origin, I'll test zero comma zero. And basically, you just plug in zero for the X and zero for the Y and see if the inequality is true. So have zero is less than five multiplied by zero squared plus three. That'll be that zero is less than zero, squared is zero and five multiplied by zero is zero. So zero is less than zero plus three, which means that zero is less than three, which is true. So that means that the origin falls within the region that we have to shade. So now we can get to graphing. So we're drawing a Y axis and the next axis and we'll draw a dashed do uh line on the Y axis where we know that or where we would mark as Y equals three because that is where our origin is, our vertexes will not be since we know in our paragraph, our vertex was at the origin zero comma zero. But we shipped three units up. That means the Y value changes three units higher. So instead of zero comma zero, we'll have zero comma three. So that's where our vertex will be. Now, as we, we stretch vertically, we'll just draw the parabola more stretched. And as we said, we will use a dashed line because this is a strict inequality. So once again, we still have the same general shape of A U. But now our vertex is at a 0.0 comma three. And since we know that the origin will be part of the shaded region, we shade underneath the parabola. So it'll include everything underneath including under the X axis. So I hope that was helpful. This is what the graph of our function will look like. Our, our inequality will look like a Parabola with a vertex at three with a dash line and the region underneath it dashed or the region underneath then shaded in. So I hope that was helpful. And until next time.

Graph each inequality. y < 3x^2 + 2

Key Concepts

Inequalities

Quadratic Functions

Graphing Inequalities

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