In Exercises 1–26, graph each inequality. y

Question

In Exercises 1–26, graph each inequality. y<x^2−1

Accepted Answer

Welcome back. I am so glad you're here. We are given this inequality. Why is less than X squared plus four and were asked to graphic, well, the first step for graphing it is I'm going to bring it down here where there's a little bit more room. And then I'm going to draw a rough sketch of a coordinate plane will have a vertical Y axis, the horizontal X axis, they come together the origin in the middle. And now for graphing and inequality, the next thing we want to do is set it to an equality. We want to set this to why equals X squared plus four. And so then we want to look at our inequality and see if our equality, our equation is included. But this is just why is less than it's not, why is less than or equal to. So when we draw our Y equals X squared plus four function line, we need to be sure that we draw a dashed line because that is not included for our inequality. The next step for graphing and inequalities, we want to look at our equation and we want to find the base function here. And the base function is a five X equals X squared. And we recall from previous lessons that F of X equals X squared is a Parabola that opens upward and has an X and Y intercept at 00. And now that we've got that base function identified, we want to look back at our equation, our equality and we want to see what transformations are occurring. The only transformation taking place here is this plus four plus four is not happening directly to the X. So that indicates a vertical shift in the vertical shift happening here because we're adding four is going to be up four units. So now instead of having a Y intercept at 00, it's going to have a Y intercept at 04, but otherwise it will have the same shape. And so we can draw that, we just need to be sure we draw dotted lines because it does not include the points on that line. Okay. Now we've got this graft and now we want to look back at our inequality and ask ourselves, where is it true that why is less than X squared plus four? So in order to figure out which areas included, we are going to choose to test points, one test point will be up here above the function line and the other test point will be from somewhere down here below the function line for the two test points. I'm going to choose one at And another at the origin at 00. Now we're going to plug in zero and 5 for the X&Y values in our inequality. So the Y value is five, we've got five is less than the X value is 00 squared plus four. Simplifying on the right hand side, zero squared is zero. So this is just four on the right hand side, is it true that five is less than four? It's not true. So we can reject the area for that test value of +05. That part's not included. Now, let's use our other test point. The y value is zero is less than the X value is zero. Again, zero squared plus four. Simplifying on the right hand side, zero squared is +00 plus four is four. So is it true that zero is less than four? Yes, it is. So because that is true, we get to shade in this whole area that includes our test point that is below the function line. And that's our graph of the inequality. Why is less than X squared plus four? Now let's bring this back up to the top and look for one that matches, we want to look for a function line that's dotted has a Y intercept at +04. It opens upward and it all be shaded underneath the function line. Okay. So looking for an intercept at 04, they all have an intercept at 04, we want one that opens upward. Well, only C and D open upward. We want one that's shaded below the line and only D is shaded below the line. So that leaves us with answer choice. D well done. We'll catch you on the next one.

In Exercises 1–26, graph each inequality. y

Key Concepts

Inequalities

Quadratic Functions

Graphing Techniques

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