Problem 41 - Precalculus with Limits

Question

In Exercises 33-46, sketch the graph (and label the vertices) of the solution set of the system of inequalities. $$\left\\{\begin{array}{l}{x > y^{2}} \\ {x< y+2}\end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system of inequalities involves the region shaded in the graph that sits between the parabola and the straight line, which is the overlap of the two separate inequalities.

Step 1: Sketch the graph of the inequality \(x > y^{2}\)

The inequality \(x > y^{2}\) represents a parabola opening to the right since \(x\) is alone on one side of the inequality. Sketch this parabola on the coordinate plane.

Step 2: Sketch the graph of the inequality \(x < y+2\)

The inequality \(x < y+2\) represents a straight line. The line will have a slope of 1 and a y-intercept of -2. This should be plotted on the same graph as the parabola from Step 1.

Step 3: Determine the solution region

The solution to the system of inequalities is the region where the inequalities overlap. This would be the region between the parabola and the straight line. This region should be shaded or filled in on the graph.

Key Concepts

Parabolic Inequalities

When it comes to graphing parabolic inequalities, we are dealing with a visual representation of all the solutions of an inequality that involves a quadratic expression. In the given exercise, the inequality \(x > y^2\) describes a set of points in the coordinate plane where the x-values are greater than the squares of their corresponding y-values. This creates a region represented by a parabola that opens to the right, which is atypical because most students are accustomed to parabolas that open upwards or downwards.

To graph this inequality, start with a regular quadratic function such as \(y = x^2\), but then 'flip' it to align with the inequality \(x = y^2\). The inequality symbol 'greater than' suggests that the solution is not on the parabola itself but rather to the right of it. Therefore, to indicate the solution set, you would shade the entire region to the right of the parabola. In this exercise, your graph should show that all points in the shaded area satisfy the original inequality.

Linear Inequalities

On the other hand, linear inequalities are simpler in concept as they generally represent solutions as a half-plane (either above or below the line) in a two-dimensional coordinate system. The inequality \(x < y + 2\) tells us that the x-values are less than the values on the line \(y + 2\). This creates a boundary line with a slope of 1 and a y-intercept of -2.

When graphing the equation of the line, use the y-intercept to mark the point (0, -2) on the y-axis, and then, because the slope is 1, you can move up one unit and over one unit to the right to find another point on the line. Connect these points to form the line. The inequality symbol 'less than' indicates that the solution area is below the line. As a result, you would shade the entire region below the line to reflect the solution set for this inequality. Together, these steps help determine the entire area that satisfies the \(x < y + 2\) condition.

Coordinate Plane

The coordinate plane is the two-dimensional surface where all of these equations and inequalities are graphically represented. It's made up of two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). The point where these axes intersect is called the origin, designated as (0, 0). Each point in this plane is defined by an ordered pair of numbers known as coordinates, typically written as (x, y), which correspond to its horizontal and vertical positions.

Graphing on the coordinate plane requires understanding how to accurately place points and draw lines or curves based on equations or inequalities. By graphing the parabolic and linear inequalities from the exercise on the same set of axes, we can visually identify the overlap between them, which represents the solution set of the system of inequalities. Remember, in this exercise, the region where the parabolic and linear inequalities intersect is the area between the curve of the parabola and the straight line. By carefully sketching and shading this region on the coordinate plane, you effectively communicate the solutions to the given system.