In Exercises 1–26, graph each inequality. y≥x^2−9

Question

Accepted Answer

Welcome back. I am so glad you're here. We have got this inequality. Why is greater than or equal to negative X squared plus five? And we are asked to graph it. So first, I'm going to give us a little bit more room to draw our graph. Next time, I'm going to draw a rough sketch of a coordinate plane will have a vertical Y axis and a horizontal X axis. They come together at the origin in the middle. Now, the first step for graphing our inequality is we're going to turn it into an equality. So we're going to write it as Y equals negative X squared plus five. And then we are going to ask ourselves is our equality included in the inequality. And we look, and we see this is greater than or equal to. So yes, it is included with the or equal to part. So when we draw our line for the function Y equals negative X squared plus five, it will be a solid line because it's included. And then the next step is that we want to look at our equality, our equation and we want to say what is the base function happening here. And in this case, the base function is where F of X equals X squared. And we recall from previous lessons that the function F of X equals X squared is a parabola facing upward and it has an X and Y intercept at 00. So there's our base function. Now we look at our Y equals negative X squared plus five. And we ask what transformations are occurring and there's two of them here. The first one is this negative in front of our X squared. That negative is not happening directly to the function. It's not happening directly to the X there because it's not inside parentheses also getting squared. So that indicates reflection over the X axis. And so instead of facing upward, our problem is going to face downward. And then the second transformation here is this plus five on the end that also is not happening directly to the X, it's not getting squared. And because it's not happening directly to the exit indicates a vertical shift. And because we are adding five, it is a vertical shift up five units. So instead of having a Y intercept at 00, there's going to be a Y intercept at 05 and then it will still curve downward and look very much the same, even though my drawing doesn't look very much the same. But there is our equation. There's our function graft on here and that's included, but we want to know what else will be true for why is greater than or equal to negative X squared plus five. And in order to figure out what else is included there, we choose to test points. One test point is going to be here above our function and the other is going to be here below our function line. So the first test point, I'd like to choose the .06 and the other function or the other test point, I would like to choose at the origin at 00. So let's plug zero and six in for our X and Y values in our inequality. So our Y value here is six. So six is greater than or equal to a negative. Our X term is 00 squared plus five. Simplifying on the right hand side, negative zero squared, that's zero. So we just have five. Is it true that six is greater than or equal to five? Yes, it is. So we get to shade in this part that is above our function line here where Y equals negative X squared plus five. Now we can test our 50.0 just to make sure we didn't make a mistake somewhere in there. So the Y value is zero is greater than or equal to negative zero again, squared plus five. The right hand side simplifies to five. So is it true that zero is greater than or equal to five? No, it's not So we know that that 50.0 is not included for our shading. Alright, so we've got this graft, we're going to bring this back up to the top. We want to look for a line that intersects the Y axis at 05 and then is shaded above the line. So let's look for that. All right. So looking at all of our answer choices, we're looking for A Y intercept at 05, they all have a Y intercept at 05. We want a Parabola that opens downward, only A and B open downward and we want the shaded portion to be above our function line and that matches with answer choice. A well done. We'll catch you on the next one.

In Exercises 1–26, graph each inequality. y≥x^2−9

Key Concepts

Inequalities

Quadratic Functions

Graphing Techniques

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