Graph each inequality. y (x - 1)^2 + 2

Question

Graph each inequality. y > (x - 1)^2 + 2

Accepted Answer

Hello, everybody. I have everything. All right. Today, today we're going to look at this question that states draw the graph for the given inequality where our inequality is Y is greater than open parentheses, X minus two, closed parentheses squared plus four. Now, the first thing I'll note is that we have a strict inequality and therefore we will have a dashed graph. So this will make more sense once we actually get to graphing. But this is something that you should know right away before you start graphing. A strict inequality is though, are those that have greater than, or less than if you had an equal in there? So let's say greater than, or equal to, or less than, or equal to? Those are non strict inequalities and you would use a solid line for your graph and you'll see kind of what that looks like better once we get there. Now, moving on, the first thing I'll do is actually replace that greater than sign with an equal sign. So we'll have Y equals a open parentheses, X minus two, closed parentheses squared plus four. And I'm only gonna do that so that it's a bit easier to solve this problem. So first, I'm going to identify our parent function. Now, the parent function is basically what function the function that our inequality is based off of. So without any transformations in our inequality, the minus two and the plus four serve as transformation. So if we take those out, we'll have a parent function and I'll label it as G of X is equal to X squared. So what do we know about X squared? Well, we know that the graph of that will be a parabola. So I'll draw a very small version of that parent function. So we'll have a Y axis and an X axis. And we know it has a vertex at zero comma zero and a point at one comma one and negative one comma one and looks like a U shaped upwards. So once again, the vertex will be at zero, comma zero and the points that we know or two points that we know are one comma one and negative one comma one. So now we have to see and add little by little the transformations of our graph and see how we go from the parent function to our current function. So first we'll do another function, let's call it H of X. And that will be equal to and let's add the first transformation. We'll just add minus two first. So we'll have open parentheses, X minus two closed parentheses squared. Now, this means this transformation will have the GEO X graphs or parent function will be shifted two units to the right. So adding that minus two inside of the parentheses, which with the X will move our pair function two units to the right. So now let's add the final transformation and that will lead us to our current function Y equals open parentheses, X minus two, closed parentheses squared plus four. Now, with that plus four, we'll have the graph of H of X that will be shifting four units up. So in total, we move our, our parent function moves two units to the right and four units upward. So let's use that information with our points. We said that we had a vertex at zero comma zero. And now we're moving that. So if we're moving two units to the right now, we have the zero plus two and if we move four units up, we'll have zero plus two comma zero plus four, which means our new vertex will be at two comma four. And now let's do that with the other points that we new. So we had a point at one comma one. And with the shifts, we now have one plus two comma one plus four, meaning our new point will be at three comma five. And we also said we had a point on negative one comma one in our paragraph. And with the shifts, we'll be at negative one plus two, comma one plus four meaning that we have a new point at one comma five. So we have a new vertex and two new points for a graph. Now because this is an inequality, we have to test a point just because we will be shading in a region. So inequalities don't have just one answer. In this case, we have, why is greater than so anything that is greater than that? Why would be an answer? So you have to shade in the region with all the answers. Now to see what region that is, you simply test any point that you want. In this case, I'll test the origin. So zero comma zero. So I'll plug in zero for the Y and zero for the X and our inequality. So we'll have that zero is greater than open parentheses, zero minus two, close parentheses squared plus four, meaning that zero is greater than zero minus two is negative two, negative two, squared is four. So zero is greater than four plus four or zero is greater than eight, which is false, meaning that the origin will not fall within our graph. So we have to shade in the region that does not include the origin. So now we're ready for graphing. So will draw a Y axis and an X axis and we can start plugging in points that we know. So let's try to make that wax as straight as possible. So we'll start plugging in points that we know I'll only label I, I know my graph will be in the first quadrant. So I only label the points on the positive side. So I'll go on the positive side of the Y axis and the positive side of the X axis. So I'll label the point. Uh I'll put a dash at 12345, six and seven. And I'll label the five in the positive side of my Y axis and or uh of my X axis. And I'll do the same thing on the positive side of my Y axis. So I'll draw a line at 1234, 567 and I'll label the, the point at five. And now I can plug in the points that I know. So I have a, I know I have a vertex at two comma four and I point at negative one comma five or um one comma five and three comma five. And now before I connect those points, I have to go back to what we said at the beginning of this question where we said that we had a strict inequality and we have a dashed graph. So instead of using a solid line to connect my points, I'll be using a dashed line to connect all of these points. And we have the same general shape as the paragraph. So a parabola and when we tested the origin, we saw that it did not fall within that region of being shaded So we shade inside of our Parabola which will exclude the origin. So the inside of that Parabola will function as the answer to all our inequality. So just like that, we are able to graph our inequality, the final shape will be a parabola with a vertex at two comma four and points at three, comma five and one comma five those points being connected by a dash line with the region inside of the Parabola being shaded. So I hope that was helpful and until next time.

Graph each inequality. y > (x - 1)^2 + 2

Key Concepts

Inequalities

Quadratic Functions

Graphing Techniques

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