In Exercises 33-46, sketch the graph (and label the vertices) of the solution set of the system

In Exercises 33-46, sketch the graph (and label the...

Key Concepts

Intersection of Regions

The intersection of regions refers to the common area shared by the solution sets of individual inequalities in a system. This concept is important because the overall solution is defined by where all the individual regions overlap, representing the set of points that satisfy every inequality simultaneously.

Linear Functions and Their Graphs

Linear functions represent straight-line relationships in the coordinate plane. Graphing linear functions involves plotting points based on the slope and intercept, which serve as boundaries for inequalities and help delineate portions of the plane that satisfy the inequality conditions.

Quadratic Functions and Their Graphs

Quadratic functions, which often appear in the form of parabolic equations, are central to understanding the shape and orientation of curves defined by second-degree expressions. Knowing how to graph a parabola, including its vertex and direction of opening, is crucial when the inequality involves quadratic terms.

Graphing Inequalities in the Coordinate Plane

Graphing inequalities entails drawing the boundary curve or line associated with each inequality and then shading the appropriate region that satisfies the inequality condition. This process helps visually determine the set of points that fulfill the given conditions.

Systems of Inequalities

A system of inequalities in two variables involves multiple inequalities that must all be satisfied simultaneously. Solving such systems graphically requires plotting the regions defined by each inequality and determining their intersection, which forms the solution set.

Transcript

00:01 Okay, so we are to graph the system of inequalities and find any points of intersection.

00:05 Okay, so you might not recognize this curve because it has a y squared.

00:11 And you might recognize it if it had an x squared.

00:13 So what i'm going to do is i'm going to replace x and y.

00:16 So set x equal to y, and then we get y minus x squared is greater than zero.

00:22 And if you solve for that, you get y is greater than x squared.

00:26 So in this case, the boundary curve, if x and y were equal to each other, and if we replace that, the boundary curve would simply be y equals x squared.

00:36 And you all know that as a parabola that looks something like this.

00:43 Okay, so that's not too bad.

00:45 Unfortunately, this isn't the same as that because i had to set x equal to y in order to get that.

00:51 So there are a couple of ways of looking at this.

00:54 The first way is to realize that by setting x equals y, the actual graph that we want is just a reflection about the line y equals x.

01:08 So if you take this curve and you flip it about this line right here, about this pole, if you want to consider it a pole, you'll get the graph that you actually want.

01:18 The second way of looking at it is since this is the y -axis and this is the x -axis, in order to get the same thing, you want the x -axis to be up here.

01:29 So you want this to be the x -axis, and you want this to be the y -axis.

01:33 Well, in order to do that, you need to take the y and flip it over here.

01:46 So if you actually flip this entire graph 90 degrees clockwise, you will get this curve, and this is exactly the curve that you want.

02:01 This is exactly x equals y squared.

02:06 Okay, so if you didn't really understand any of that, just plug in a bunch of points and you'll see that this is the exact same.

02:15 It's difficult because this isn't technically a function because it fails the vertical line test pretty much everywhere.

02:21 But that's the graph that we're looking for.

02:23 So i'm going to go ahead and try to erase all the other mess that i've created.

02:33 So that is the graph of x equals y squared.

02:39 That's the boundary curve.

02:40 Now, to figure out, now since this is greater than and not equal to, this isn't going to be a solid curve.

02:47 Dashed, so i have to get rid of that.

02:50 And i have to find a region to shade...

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