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Chapter 12: Problem 31

Graph each system of inequalities. \(x^{2}+y^{2}>9\) \(y>x^{2}-1\)

Short Answer

Expert verified
The solution is the region outside the circle of radius 3 and above the parabola \(y = x^2 - 1\).

Step by step solution

01

Understand the Inequalities

The system includes two inequalities: 1. The region outside the circle defined by the equation \(x^2 + y^2 > 9\).2. The region above the parabola defined by the equation \(y > x^2 - 1\).
02

Graph the Circle

Draw the circle \(x^2 + y^2 = 9\) with the center at the origin \((0, 0)\) and radius 3. Shade the region outside the circle because we have \(x^2 + y^2 > 9\).
03

Graph the Parabola

Draw the parabola \(y = x^2 - 1\). It is a standard parabola that opens upwards, starting from the point \((0, -1)\). Shade the region above the parabola because we have \(y > x^2 - 1\).
04

Determine the Overlap Region

The solution to the system of inequalities is the region that satisfies both conditions: outside the circle and above the parabola. Shade the overlap of these two regions.

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Key Concepts

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

Understanding Inequalities
Inequalities in mathematics show the relationship between two expressions and how one is larger or smaller compared to the other. They are graphically represented in coordinate planes to provide visual solutions to problems. For our given problem, we have two inequalities:
  • For the circle: \(x^2 + y^2 > 9\) signifies all points outside the circle.
  • For the parabola: \(y > x^2 - 1\) signifies all points above the parabola.
It鈥檚 essential to understand what regions these inequalities represent before graphing.
Graphing Circles
A circle in the coordinate plane is graphically represented by its equation. The standard form of a circle's equation is \(x^2 + y^2 = r^2\), which has a center at \((0,0)\) and a radius of \(r\). Given our first inequality \(x^2 + y^2 > 9\):
  • We first draw the circle \(x^2 + y^2 = 9\), where the center is at the origin and the radius is 3 (since \(9 = 3^2\)).
  • Since the inequality is \(>\), we shade the region outside this circle. This is because \(>\) indicates all points at a distance greater than 3 from the origin.
Visualizing the circle helps identify and understand where the inequality holds true and aids in graphing complex systems involving circles.
Graphing Parabolas
A parabola is a U-shaped curve represented by the equation \(y = ax^2 + bx + c\). It opens upwards if \(a > 0\) and downwards if \(a < 0\). For our inequality \(y > x^2 - 1\) we follow these steps:
  • First, we sketch the parabola \(y = x^2 - 1\).
  • This parabola opens upwards because the coefficient of \(x^2\) is positive. It starts at the vertex point \((0, -1)\) and extends up.
  • Since the inequality is \(>\), we shade the region above this parabola where all \(y\) values are greater than the parabola鈥檚 values at each \(x\)-coordinate.
Combining this graph with other inequalities provides a unified visual solution where regions overlap, making graphing inequalities involving parabolas straightforward.

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Most popular questions from this chapter

Which one of the following is a description of the graph of the solution set of the following system? \(x^{2}+y^{2}<25\) \(y>-2\) A. All points outside the circle \(x^{2}+y^{2}=25\) and above the line \(y=-2\) B. All points outside the circle \(x^{2}+y^{2}=25\) and below the line \(y=-2\) C. All points inside the circle \(x^{2}+y^{2}=25\) and above the line \(y=-2\) D. All points inside the circle \(x^{2}+y^{2}=25\) and below the line \(y=-2\)

Suppose that a nonlinear system is composed of equations whose graphs are those described, and the number of points of intersection of the two graphs is as given. Make a sketch satisfying these conditions. (There may be more than one way to do this.) A line and a circle; one point

Solve each system using the substitution method. \(y=x^{2}+8 x+16\) \(x-y=-4\)

Graph each hyperbola with center shifted away from the origin. $$ \frac{(x+3)^{2}}{16}-\frac{(y-2)^{2}}{25}=1 $$

Graph each system of inequalities. \(3 x-4 y \geq 12\) \(x+3 y>6\) \(y \leq 2\)
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