/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 19 Sketch the graph of each nonline... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 19

Sketch the graph of each nonlinear inequality. $$y>-x^{2}$$

Short Answer

Expert verified
Sketch a downward-opening parabola with a dashed line at $$y = -x^2$$ and shade the region above it.

Step by step solution

01

Understand the inequality

The given inequality is $$y > -x^2$$. This inequality describes a region above the graph of the equation $$y = -x^2$$. The graph of $$y = -x^2$$ is a parabola that opens downwards.
02

Sketch the parabola

Draw the graph of the parabola described by the equation $$y = -x^2$$. This is a standard parabola opening downwards with its vertex at the origin (0, 0). Plot several points (such as (1, -1), (-1, -1), (2, -4), (-2, -4)) and connect them with a smooth curve.
03

Identify the solution region

Since the inequality is $$y > -x^2$$, we want the region above the parabola. The parabola itself is included as a boundary, but it is not part of the solution because the inequality is strict (>). Therefore, draw the parabola as a dashed line to indicate that points on the parabola are not included in the solution.
04

Shade the solution region

Shade the region above the dashed parabola to represent all the points where $$y > -x^2$$. This shaded region is the solution to the inequality.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sketching Graphs
When working with nonlinear inequalities like the one given in the exercise, the first step is to understand the underlying graph. In this case, we have the inequality \(y > -x^2\). To sketch the graph, start by drawing the related equation \(y = -x^2\). This is a simple parabola that opens downwards, with its vertex at the origin (0, 0). Plotting several key points helps in visualizing the curve more accurately. For instance, plot points like (1, -1), (-1, -1), (2, -4), and (-2, -4). Connect these points with a smooth curve to complete your graph. Remember, the goal is to use this sketch to help visualize and solve the inequality.
Nonlinear Inequalities
A nonlinear inequality, such as \(y > -x^2\), describes a region in the coordinate plane rather than just a line or curve. In our case, the inequality indicates that we are looking for all the points where the value of \(y\) is greater than the value of \(-x^2\). To determine which area this is, we first sketch the boundary set by the related equation, \(y = -x^2\). Because the inequality is strict (>), this boundary is represented as a dashed line. It signifies that the points on the parabola itself are not included in the solution region. Drawing the boundary correctly is essential as it directly influences the solution region.
Solution Region
The solution region for the inequality \(y > -x^2\) is found above the parabola we have sketched. The area above the dashed parabola represents all the points that satisfy the inequality, meaning for every \(x\) in that region, the corresponding \(y\) value will be greater than \(-x^2\). Shading this region on your graph conveys the solution visually. It鈥檚 important to ensure that the parabola is dashed and not solid, to make it clear that points on the parabola itself are not part of the solution. By following these steps鈥攕ketching the graph, identifying the boundary, and shading the appropriate region鈥攜ou can effectively solve and visualize nonlinear inequalities.

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