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 and Vertex Analysis

When graphing systems of inequalities, determining the vertices of the resulting region often involves finding the points of intersection between the boundary curves. These vertices are critical in accurately sketching the solution area because they define the limits and shape of the overlapping regions.

Quadratic Inequalities and Parabolic Functions

Quadratic inequalities, such as those involving parabolic equations, require understanding the shape and direction of the parabola. In these cases, analyzing the vertex, axis of symmetry, and intercepts helps determine the portion of the graph that satisfies the inequality. This is a key skill in representing the regions defined by quadratic functions.

Graphing Systems of Inequalities

This concept involves representing multiple inequalities on a common coordinate system, where each inequality defines a region. The overall solution is the intersection of these regions, meaning the area that satisfies every given inequality. Understanding how to graph each inequality by drawing its boundary and shading the appropriate side is essential to determining the solution set.

Boundary Curves and Regions

The boundaries in graphing inequalities are defined by replacing the inequality with an equality, which creates curves or lines. These boundaries demarcate the regions of interest. It is important to know whether the boundary is included (closed) or excluded (open) based on the inequality, as this affects how the solution set is represented graphically.

Transcript

00:01 To graph the system of inequalities and label any points of intersection.

00:04 So the first one looks a little odd because we have a y squared, but it wouldn't look odd if we replaced x with y.

00:12 So if x equals y, this inequality would become y is less than x minus x squared.

00:20 So in this instance, the boundary curve would simply be y is equal to x minus x squared.

00:26 And this is a parabola that we can hopefully easily graph.

00:32 Okay, so how would we graph this? well, if we factor an x out, we would get that y would be equal to zero.

00:46 In other words, the points of intersection, the x intercepts, would be 0 and 1.

00:57 And since this is a downward shaping parabola, because the number in front of x squared is negative, it's going to look like this.

01:10 So the vertex is going to be somewhere above the x -axis.

01:16 And because the parabola is symmetric, it's actually going to be right between 0 and 1.

01:22 So it's going to be here at a half.

01:25 So if i plug in a half as my x value, i get a half minus a half squared, which is a fourth.

01:38 So that's going to be a fourth.

01:42 So if this is one, it's really just going to be here.

01:48 So that's my parabola.

01:51 Or that's at least what my parabola would be if x equal y.

01:57 Okay, but you'll find that really this is kind of the same thing.

02:01 And in fact if you turn this 90 degrees clockwise you flip it 90 degrees you'll get the same exact thing okay so if i plug in if i flip x and y in order to now graph the actual boundary curve which is this you'll find the exact same procedure right so instead of x intercepts we'll find that the y intercepts are going to be 0 and 1 all right and it's again going to be a downward kind of facing parabola with a vertex that's going to be a fourth right here.

02:50 So this is located at a fourth.

02:52 I know it doesn't look like a fourth, but just take my word for it.

02:56 And the parabola is just going to be like that.

03:03 So that's exactly what you get if you took the red curve and you flipped it 90 degrees.

03:17 Okay...

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