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Chapter 7: Problem 21

Graph inequality. \(y \geq x^{2}-9\)

Short Answer

Expert verified
The graph of the inequality \(y \geq x^{2} - 9\) is the graph of the parabola \(y = x^{2} - 9\), with all the area above this graph shaded to signify y-values greater than the value of the function.

Step by step solution

01

Understand the function

This is the equation of a parabola that opens upwards. The graph of the function \(y = x^{2} - 9\) is a parabola that opens upwards with the vertex at the point (0,-9).
02

Sketch the graph

First, sketch the graph of the parabola \(y = x^{2} - 9\), with a vertex at the point (0, -9). The graph will open upwards.
03

Determine the shading area

The inequality states that \(y \geq x^{2} - 9\). This means y is greater than or equal to \(x^{2} - 9\). So, shade the region above the parabola to represent all the points (x,y) where the y-values are greater than the value of the parabola.

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Key Concepts

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

Parabolas
Parabolas are U-shaped graphs that can have either an upward or downward opening. The basic equation of a parabola is given by \(y = ax^{2} + bx + c\), where \(a\), \(b\), and \(c\) are constants. The parabola in this exercise, \(y = x^{2} - 9\), has an "a" value of 1, which means it opens upwards since positive values of "a" create upward-opening graphs. Understanding the direction the parabola opens is crucial in graphing because it affects the region where inequality solutions exist. In this exercise, the graph opens upwards, forming a U-shape on the coordinate plane.
Vertex
The vertex of a parabola is a significant point because it represents either the minimum or maximum point of the curve, depending on its opening direction. For an upward-opening parabola like \(y = x^{2} - 9\), the vertex is the lowest point on the graph. The vertex's coordinates can often be found directly from the equation in vertex form \(y = a(x-h)^{2} + k\), where \((h, k)\) is the vertex. Since \(y = x^{2} - 9\) can be rewritten as \(y = (x-0)^{2} - 9\), we identify the vertex as \((0, -9)\). Knowing the vertex helps in accurately sketching the parabola on a graph.
Shading Regions
When graphing inequalities such as \(y \geq x^{2} - 9\), shading is used to represent the solutions that satisfy the inequality. After drawing the parabola, the next step is to determine which region of the graph to shade. The inequality \(y \geq x^{2} - 9\) tells us that we need values of \(y\) that are equal to or greater than those on the parabola. This translates to filling in the area above the curve, including the curve itself since the inequality uses \(\geq\). Shading makes it visually clear which points on the graph satisfy the inequality, highlighting the set of all possible solutions.

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Most popular questions from this chapter

Solve each system by the method of your choice. $$\left\\{\begin{array}{l} \frac{2}{x^{2}}+\frac{1}{y^{2}}=11 \\ \frac{4}{x^{2}}-\frac{2}{y^{2}}=-14 \end{array}\right.$$

Make a rough sketch in a rectangular coordinate system of the graphs representing the equations in each system. The system, whose graphs are a line with positive slope and a parabola whose equation has a positive leading coefficient, has two solutions.

Compare the graphs of \(3 x-2 y>6\) and \(3 x-2 y \leq 6\) Discuss similarities and differences between the graphs.

Let \(x\) represent one number and let \(y\) represent the other number. Use the given conditions to write a system of nonlinear equations. Solve the system and find the numbers. The sum of two numbers is 20 and their product is \(96 .\) Find the numbers.

Solve each system by the method of your choice. $$\left\\{\begin{array}{l} x^{3}+y=0 \\ x^{2}-y=0 \end{array}\right.$$
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