In Exercises 39–45, graph each inequality. y ≤ x^2 - 1

Question

Accepted Answer

Hey everyone welcome back in this problem. We are asked to draw the graph of the given inequality in the rectangular coordinate system were given why is greater than or equal to four X squared plus one. Now, when it talks about the rectangular coordinate system, if you look at the graphs they've given you for potential potential answers. What you'll see is that the X axis, the markings along the X axis are bigger than along the y axis. Okay. And so the boxes look like rectangles. Okay, so that's all that means by the rectangular coordinate system, one unit along the X axis is just drawn bigger than along the y axis. Alright, so let's scroll down, give ourselves some room to work and we're just going to rewrite the inequality we were given. So we can refer to it. We were given why is bigger than or equal to four X squared plus one. Now when we're given an inequality like this and asked to graph it, the first thing we wanna do is graph the equality. Okay, so we want to graph y is equal to four X squared plus one. And what we'll see here is because we have greater than or equal to this equality is going to be included. Alright, so we look at this equation and this is a quadratic. Okay, so let's figure out the X and Y intercept that's just going to help us with our graph. So let's start with X intercepts. Okay, that's gonna be one Y zero. We get four X squared plus one. Okay, this is gonna tell us that X squared is equal to a negative 1/4. Okay. And so what we see here is that we have no real roots. Okay because we have to find the route, we have to take the square root, we have the square root of a negative so we have no real roots. Okay so this function is not going to touch across the X axis. Now let's do the same for the Y intercept we get y is equal to one. Ok. That y intercept is one X zero. So we have y equals one. So we have the 10.1. And what you'll also notice is that this coefficient is positive? Okay the leading coefficient is positive. So this quadratic is going to open upwards. We're gonna draw our axes and again rectangular coordinates. So the ticks along the X. Are going to be more spaced out then. Those along the Y axis, its axis Y axis. Whoops. Not Y axis negative X axis. Sorry access up here. There we go. Okay So drawing this we know we have the .01. We know this opens upwards. Okay and so we're just gonna have a parabola that looks like something like this. Okay, that's not the best drawing. It should be let me try try again because it should be symmetrical about that axis about the Y axis. Okay that's a little bit better. Alright so we have this problem here. Now we need to figure out what part of this plane is included. Okay, let me draw this Parabola up a little bit higher. And what you'll notice is that this Parabola cuts the plane into two portions. Okay, There's this part kind of inside of the parabola above it and then the rest of the plane that's kind of below it or outside of it. We want to figure out which one of those is included in our inequality. Well, we have Y. Is greater than or equal to four X squared plus one. That means that the part of our inequality that's included is going to be above this problem. Okay, Greater than this Parabola. So we are above this problem. All right. Now, if we were confused or if that doesn't make sense and we want to figure out what part of this plane was included in what parts not what you can do is choose a particular point. Okay. So say you choose the zero. If you put this into the inequality, if you're going to get zero is greater than or equal to one. Okay, well that's not true. So that tells us that the zero is not included. Okay, so we know that that portion of the plane that includes 00 is not included. That means the other portion is And you could do the same thing. You could pick a point within this blue part that we've drawn and you'll find that it does satisfy the inequality. Okay. Alright. So that's our graph of the inequality. Let's go back up to our answer choices. All right. Now you'll see as B and D. They look very similar. What's different is where the Y intercept is. Okay. And B. We have X intercepts were crossing in our Y intercept is negative. That's not the case. We have right or Y intercept is at one. We don't cross that X axis. And so what we have is answered. Alright, thanks everyone for watching. I hope this video helped you in the next one.